%% @texfile{ %% filename="Deform.tex", %% version="1.1", %% date="Sep-1996", %% cdate="19960908", %% filetype="AMSTeX", %% journal="J. d'Analyse Math. 70, 267-324 (1996)", %% doi="10.1007/BF02820446", %% copyright="Springer". %% } \input amstex \loadbold \documentstyle{amsppt} \magnification=1200 \pagewidth{14cm} \baselineskip=15 pt \NoBlackBoxes \TagsOnRight \font\myfont=cmsy10 scaled \magstep3 \def\arrowl{\text{\myfont \char"00\hskip-1.2mm\char"21}} \def\arrowse{\text{\myfont \&}} \def\arrowne{\text{\myfont \%}} \def\smat#1#2{\hskip -3mm\spreadmatrixlines{-2mm}\matrix \\\endmatrix} \def\tsmat#1#2{\hskip -4mm\matrix &{\ssize#1}\\ \vspace{-2mm} &{\ssize#2}\endmatrix} \def\subjclass#1{MSC: #1} \def\keywords#1{Keywords: #1} \def\gap{\vskip 0.1in\noindent} \def\ref#1#2#3#4#5#6{#1, {\it #2,} #3 {\bf #4} (#5), #6.} %References \def \baum {1} % Baumgartner \def\belok {2} % Belokolos \def\bggss {3} % Bolle et al. \def\bufi {4} % Buys-Finkel \def\crum {5} % Crum \def\dar {6} % Darboux \def\deift {7} % Deift \def\detr {8} % Deift-Trubowitz \def\eastkalf {9} % Eastham-Kalf \def\ehkn {10} % Ehlers-Knorrer \def\erfl {11} % Ercolani-Flaschka \def\fit {12} % Finkel-Isaacson-Trubowitz \def\fir {13} % Firsova \def\flmcl {14} % Flaschka-McLauglin \def\ggkm {15} % Gardner et al. \def\gellev {16} % Gel'fand-Levitan \def\gdekker {17} % Gesztesy Dekker \def\gcam {18} % Gesztesy Cambridge \def\gjfa {19} % Gesztesy JFA \def\gkt {20} % Gesztesy-Krishna-Teschl \def\gnow {21} % Gesztesy-Nowell-Potz \def\gss {22} % Gesztesy-Schweiger-Simon \def\gstrans {23} % Gesztesy-Simon Trans. \def\gsacta {24} % Gesztesy-Simon Acta \def\gst {25} % Gesztesy-Simon-Teschl \def\gsvi {26} % Gesztesy-Svirsky \def\gtproc {27} % Gesztesy-Teschl Proc. \def\gtdiff {28} % Gesztesy-Teschl Diff. \def\gwei {29} % Gesztesy-Weikard \def\gzh {30} % Gesztesy-Zhao \def\grma {31} % Grosse-Martin \def\iwa {32} % Iwasaki \def\jac {33} % Jacobi \def\kaymo {34} % Kay-Moses \def\kumi {35} % Kuznetsov-Mikhailov \def\lei {36} % Leighton \def\levbook {37} % Levitan book \def\levsbor {38} % Levitan Sbornik \def\mar {39} % Marchenko \def\mckcpam {40} % McKean CPAM \def\mckrev {41} % McKean Revista \def\mckjsp {42} % McKean JSP \def\mckspm {43} % McKean Symp. Pure Math. \def\mcknew {44} % McKean CPAM 92 \def\mcktr {45} % McKean-Trubowitz \def\mckmoe {46} % McKean-van Moerbeke \def\nmpz {47} % Novikov et al. \def\potr {48} % Poschel-Trubowitz \def\ratr {49} % Ralston-Trubowitz \def\schm {50} % Schmincke \def\simon {51} % Simon \def\te {52} % Teschl \def\tpre {53} % Teschl preprint \topmatter \title Spectral Deformations of One-Dimensional Schr\"odinger Operators \endtitle \rightheadtext{Spectral Deformations of One-Dimensional Schr\"odinger Operators} \author F.~Gesztesy$^1$, B.~Simon$^2$, and G.~Teschl$^3$ \endauthor \leftheadtext{F.~Gesztesy, B.~Simon, and G.~Teschl} \thanks$^1$ Department of Mathematics, University of Missouri, Columbia, MO 65211. E-mail: gesztesyf\@missouri.edu \endthanks \thanks$^2$ Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA 91125. E-Mail: bsimon\@caltech.edu. This material is based upon work supported by the National Science Foundation under Grant No.~DMS-9401491. The Government has certain rights in this material. \endthanks \thanks$^3$ Institut f\"ur Reine und Angewandte Mathematik, RWTH Aachen, 52056 Aachen, Germany. Current address: Institut f\"ur Mathematik, Strudlhofgasse 4, 1090 Wien, Austria. E-mail: gerald.teschl\@univie.ac.at \endthanks \thanks {\it{J.~d'Analyse Math. {\bf 70}, 267--324 (1996)}} \endthanks \thanks \subjclass{Primary 34B24, 34L05; Secondary 34B20, 47A10} \endthanks \thanks \keywords{Spectral deformations, Sturm-Liouville operators, isospectral} \endthanks \date June 18, 1996 \enddate \abstract We provide a complete spectral characterization of a new method of constructing isospectral (in fact, unitary) deformations of general Schr\"odinger operators $H=-\frac{d^2}{dx^2}+V$ in $L^2 (\Bbb R)$. Our technique is connected to Dirichlet data, that is, the spectrum of the operator $H^D$ on $L^2 ((-\infty, x_0))\oplus L^2 ((x_0, \infty))$ with a Dirichlet boundary condition at $x_0$. The transformation moves a single eigenvalue of $H^D$ and perhaps flips which side of $x_0$ the eigenvalue lives. On the remainder of the spectrum, the transformation is realized by a unitary operator. For cases such as $V(x)\to \infty$ as $|x|\to\infty$, where $V$ is uniquely determined by the spectrum of $H$ and the Dirichlet data, our result implies that the specific Dirichlet data allowed are determined only by the asymptotics as $E\to\infty$. \endabstract \endtopmatter \document \flushpar{\bf \S 1. Introduction} \vskip 0.1in Spectral deformations of Schr\"odinger operators in $L^2 (\Bbb R)$, isospectral and certain classes of non-isospectral ones, have attracted a lot of interest over the past three decades due to their prominent role in connection with a variety of topics, including the Korteweg-de Vries (KdV) hierarchy, inverse spectral problems, supersymmetric quantum mechanical models, level comparison theorems, etc. In fact, the construction of $N$-soliton solutions of the KdV hierarchy (and more generally, the construction of solitons relative to reflectionless backgrounds) is a typical example of a non-isospectral deformation of $H=-\frac{d^2}{dx^2}$ in $L^2 (\Bbb R)$ since the resulting deformation $\tilde H = -\frac{d^2}{dx^2} + \tilde V$ acquires an additional point spectrum $\{\lambda_1, \dots, \lambda_N\}\subset (-\infty, 0)$ such that $$ \sigma(\tilde H) =\sigma (H)\cup \{\lambda_1, \dots, \lambda_N\} $$ ($\sigma(\,\cdot\,)$ abbreviating the spectrum). On the other hand, the solution of the inverse periodic problem and the corresponding solution of the algebro-geometric quasi-periodic finite-gap inverse problem for the KdV hierarchy (and certain almost-periodic limiting situations thereof) are intimately connected with isospectral (in fact, unitary) deformations of a given base (background) operator $H=-\frac{d^2}{dx^2}+V$\!. Although not a complete bibliography on applications of spectral deformations in mathematical physics, the interested reader may consult [\baum], [\belok], [\bggss], [\gjfa], [\grma], [\nmpz], and the references cited therein. Our main motivation in writing this paper descends from our interest in inverse spectral problems. As pointed out later (see Remarks 4.5, 4.7, and 4.8), spectral deformation methods can provide crucial insights into the isospectral class of a given base potential $V$\!, and in some cases can even determine the whole class $\text{Iso}(V) =\{\tilde V\in L^1_{\text{\rom{loc}}} (\Bbb R)\mid\sigma (-\frac{d^2}{dx^2} +\tilde V)=\sigma (-\frac{d^2}{dx^2}+V)\}$ of such potentials. A particularly ``annoying" open problem in inverse spectral theory concerns the characterization of the isospectral class of potentials $V$ with purely discrete spectra (e.g., the harmonic oscillator $V(x)=x^2$). In [\gsacta], we proposed a way to label the isospectral operators for such a situation with Dirichlet data. Fix $x_0$ and let $H^D$ be the operator in $L^2 ((-\infty , x_0)) \oplus L^2 ((x_0, \infty))$ with Dirichlet boundary condition at $x_0$. $H^D = H^D_- \oplus H^D_+$. The Dirichlet data are the pairs $(\mu, \sigma)$ with $\mu\in\Bbb R$, $\sigma\in \{+, -\}$ of eigenvalues of $H^D$ and a label of whether they are eigenvalues of $H^D_-$ or $H^D_+$. We showed in [\gstrans], Theorem 3.6, that for any Dirichlet data, there is at most one $V$ in the isospectral class of a given $-\frac{d^2}{dx^2}+V_0$ with discrete spectrum so that $V$ has the given Dirichlet data (in the degenerate case, where any eigenvalues of $H^D$ and $H$ coalesce, one must include an additional parameter in the Dirichlet data for each coincidence of eigenvalues, see Remarks 4.9 and 4.10). That is, the map from $V$ to Dirichlet data is one-one when defined on the isospectral set of potentials. The issue is determining the range of this map. While this paper does not solve the inverse discrete spectral problem, it will make one important contribution. As a result of our principal Theorem 4.4, we obtain the fact that for any potential $V$\!, any finite number of deformations of Dirichlet data (i.e., Dirichlet eigenvalues together with their left/right half-line distribution, see (2.7)) in spectral gaps of $V$ produce isospectral deformations $\tilde V\in \text{Iso}(V)$ of $V$\!. In particular, there are no further constraints on these Dirichlet data (except, of course, these deformations are required to be finite in number and to stay within the spectral gaps in question). Applied to the inverse discrete spectral problem, this means that any constraints enforced on Dirichlet data can only be asymptotic in nature, that is, can only come from their ``tail end" at infinity. That such asymptotic constraints necessarily exist is a consequence of a recently proved general trace formula for $V(x)$ [\gsacta] (see Remark 4.8). The precise nature of these constraints, however, is unknown to date. Mathematically, the techniques involved to produce isospectral $\tilde V$ or classes of non-isospectral ones where eigenvalues are added or removed, but the remaining spectral characteristics stay identically to those of the base potential $V$\!, can be traced back to commutation methods. These commutation methods in turn are intimately connected with factorizations of the Schr\"odinger differential expression $-\frac{d^2}{dx^2}+V(x)$ into products of first-order differential expressions. More precisely, one seeks a factorization of the type $$\gather -\frac{d^2}{dx^2} +V(x) =\alpha(\lambda) \alpha(\lambda)^+ + \lambda, \\ \alpha(\lambda) =\frac{d}{dx} +\phi (\lambda, x), \quad \alpha(\lambda)^+ = -\frac{d}{dx} +\phi(\lambda, x) \endgather $$ for some appropriate $\lambda\in\Bbb R$. A subsequent commutation of the factors $\alpha (\lambda)$ and $\alpha (\lambda)^+$, introducing the differential expression $$ -\frac{d^2}{dx^2} + \tilde V(\lambda, x) =\alpha (\lambda)^+ \alpha(\lambda) +\lambda, $$ then yields associated isospectral or special classes of non-isospectral deformations $\tilde V(\lambda, x)$ of $V(x)$ depending on the choice of $\phi (\lambda, x)$ in $\alpha(\lambda), \alpha(\lambda)^+$. In the following, we briefly outline three different instances of commutation techniques that occur in the literature. We start with the single commutation or Crum-Darboux method (going back at least to Jacobi). In this method, $H=-\frac{d^2}{dx^2}+V$ is assumed to be bounded from below, $\inf\sigma(H) >-\infty$, and $\lambda\in\Bbb R$ is chosen according to $\lambda <\inf\sigma(H)$. One sets $\phi (\lambda, x)=\psi'_\nu (\lambda, x)/\psi_\nu (\lambda, x)$, where $\psi_\nu$ satisfies $\psi'' = (V-\lambda)\psi$ (cf.~(A.7)). Depending on the choice of $\psi_\nu (\lambda, x)$, the aforementioned commutation procedure yields a spectral deformation $\tilde H_\nu (\lambda)$ of $H$, $$ \tilde H_\nu (\lambda) =-\frac{d^2}{dx^2} + \tilde V_\nu (\lambda, x), \quad \tilde V_\nu (\lambda,x) = V(x) -2\{\ln [\psi_\nu (\lambda, x)]\}'', \quad x\in\Bbb R, \ \lambda < \inf\sigma(H), $$ which is either isospectral to $H$ or acquires the additional eigenvalue $\lambda$ below the spectrum of $H$, that is, $$ \text{either } \sigma(\tilde H_\nu (\lambda)) =\sigma(H) \text{ or } \sigma(\tilde H_\nu (\lambda)) =\sigma(H)\cup\{\lambda\}. $$ Moreover, it can be proved that the remaining spectral characteristics of $H$ remain preserved in the sense that $\tilde H_\nu (\lambda)$ and $H$, restricted to the orthogonal complement of the eigenspace associated with $\lambda$, are unitarily equivalent. A summary of this technique, as well as pertinent references to its extraordinary history and to more recent applications of it, will be given in Appendix A. The fact that $\lambda$ is required to lie below the spectrum of $H$ is clearly a severe limitation. One possibility to avoid this restriction is provided by the following second technique, the double commutation method. Formally, this method can be obtained from two successive single commutations at a point $\lambda\in\Bbb R\backslash\sigma_{\text{\rom{ess}}}(H)$ ($\sigma_{\text{\rom{ess}}}(\,\cdot\,)$ the essential spectrum), or equivalently, as the result of two single commutations at $\lambda'$ and $\lambda''$, $\lambda'\neq \lambda''$, $\lambda', \lambda''\in\Bbb R\backslash \sigma_{\text{\rom{ess}}}(H)$ with a subsequent limiting procedure $\lambda' \to \lambda$ and $\lambda''\to\lambda$. The final outcome can be sketched as follows. Pick $\gamma >0$, $\lambda\in\Bbb R\backslash \sigma_{\text{\rom{ess}}}(H)$ and real-valued $\psi_\pm (\lambda, \,\cdot\,) \in L^2 ((R, \pm\infty))$, $R\in\Bbb R$ satisfying $\psi'' (\lambda) = (V-\lambda) \psi (\lambda)$. The spectral deformation $\tilde H_{\pm, \gamma}(\lambda)$ of $H$ is then given by $$\align \tilde H_{\pm, \gamma}(\lambda) &=-\frac{d^2}{dx^2} + \tilde V_{\pm, \gamma} (\lambda, x), \\ \tilde V_{\pm, \gamma}(\lambda, x) &= V(x) -2\biggl\{\ln \biggl[ 1\mp\gamma \int\limits^x_{\pm\infty} dx'\, \psi_\pm (\lambda, x')^2\biggr]\biggr\}'', \quad \gamma >0, \ \lambda\in\Bbb R\backslash \sigma_{\text{\rom{ess}}}(H). \endalign $$ In this case, one can show that $$ \sigma (\tilde H_{\pm, \gamma}(\lambda)) =\sigma (H) \cup \{\lambda\} $$ and again $\tilde H_{\pm, \gamma}(\lambda)$ and $H$ are unitarily equivalent upon restriction onto the orthogonal complements of their eigenspaces corresponding to $\lambda$. A summary of this method together with appropriate references to its history, as well as to recent applications of it, will be provided in Appendix B. Finally, and most importantly in connection with the contents of this paper, we shall describe a third commutation method first introduced by Finkel, Isaacson, and Trubowitz [\fit] in 1987 in connection with an explicit realization of the isospectral torus of periodic potentials. This method was again used by Buys and Finkel [\bufi] (see also Iwasaki [\iwa]) in the context of periodic finite-gap potentials and by P\"oschel and Trubowitz [\potr] and Ralston and Trubowitz [\ratr] for various boundary value problems on compact intervals. As in the previous case, this method formally consists of two single commutations, but this time at different values of the spectral parameter. The principal contribution of this paper is a generalization of the work of Finkel, Isaacson, and Trubowitz to arbitrary (i.e., not necessarily periodic) base potentials $V(x)$ and a complete spectral characterization of this commutation technique. As a result we obtain a powerful new tool in constructing sets of isospectral potentials for arbitrary base potentials. We briefly sketch this approach. Suppose $\psi_\pm (z, \,\cdot\,)\in L^2 ((R, \pm\infty))$, $z\in\Bbb C\backslash\sigma_{\text{\rom{ess}}}(H)$, $R\in\Bbb R$ satisfy $\psi''(z)= (V-z)\psi(z)$, and pick a spectral gap $(E_0, E_1)$ of $H$ with $\mu, \tilde\mu\in (E_0, E_1)$. Define $$ W_{(\tilde\mu, \tilde\sigma)} (x) =(\tilde\mu -\mu)^{-1} W(\psi_\sigma(\mu), \psi_{-\tilde\sigma}(\tilde\mu))(x), \quad \sigma, \tilde\sigma\in \{-,+\}, \ x\in\Bbb R, $$ where $W(f,g)(x)=f(x)g'(x) -f'(x)g(x)$ denotes the Wronskian of $f$ and $g$ (taking limits if $\tilde\mu =\mu$). The spectral deformation $\tilde H_{(\tilde\mu, \tilde\sigma)}$ of $H$ is then given by $$ \tilde H_{(\tilde\mu, \tilde\sigma)} =-\frac{d^2}{dx^2} + \tilde V_{(\tilde\mu, \tilde\sigma)}, \quad \tilde V_{(\tilde\mu, \tilde\sigma)}(x)=V(x)-2\{\ln [W_{(\tilde\mu, \tilde\sigma)} (x)]\}''. $$ In order to define $\tilde V_{(\tilde\mu, \tilde\sigma)}$, one needs, of course, to show that $W_{(\tilde\mu, \tilde\sigma)}(x)$ is non-vanishing on $\Bbb R$. Indeed, the key to our extension of this method to the whole line is precisely our proof in [\gst] that this Wronskian is non-zero. This proof avoids the indirect argument of [\fit], [\potr], [\ratr] that relies on compactness of the underlying interval. (Even if one is only interested in the compact interval case, our direct proof is simpler than their indirect argument.) In addition to allowing the extension to whole line problems in principle, this paper provides explicit calculations in the change of Weyl-Titchmarsh and spectral functions. In our main result, Theorem 4.4, we shall prove that $$ \sigma(\tilde H_{(\tilde\mu, \tilde\sigma)}) = \sigma(H), $$ in fact, $\tilde H_{(\tilde\mu, \tilde\sigma)}$ and $H$ will turn out to be unitarily equivalent. Moreover, if $(\mu,\sigma)$ is a Dirichlet datum for $H$ with respect to the reference point $x_0$, then all Dirichlet data for $\tilde H_{(\tilde\mu, \tilde\sigma)}$ with respect to $x_0$ are identical to those of $H$, except that $(\mu,\sigma)$ is removed and $(\tilde\mu, \tilde\sigma)$ is added instead. These results and a variety of extensions thereof constitute the principal new material in this paper. Because the spectral types of all operators in [\fit], [\potr], and [\ratr] are explicitly known, the unitarity theorem is a trivial consequence of the determination of spectra. However, for general base potentials, the spectral types can be exotic so that the unitarity result is much stronger than a mere equality result of spectra. Our proof of the unitarity relies on the explicit formula of the changes in the spectral matrix. Section 2 provides the background needed in the remainder of this paper. Section~3 treats Weyl-Titchmarsh $m$-functions and spectral functions associated with half-line Dirichlet operators. Section 4 contains our principal results on isospectral deformations and provides a complete spectral characterization of this deformation method. In particular, the Weyl-Titchmarsh $M$-matrix and spectral matrix of the deformation $\tilde H_{(\tilde\mu, \tilde\sigma)}$ are computed in terms of the corresponding matrices of the base operator $H$. A variety of additional results and possible extensions, including limit point/limit circle considerations, iterations of isospectral deformations, general Sturm-Liouville operators on arbitrary intervals, and scattering theory, are treated in Section 5. Finally, the single and double commutation methods are reviewed in Appendices A and B, respectively. \vskip 0.3in \flushpar{\bf \S 2. Preliminaries on the Dirichlet Deformation Method} \vskip 0.1in This section sets the stage for a complete spectral characterization of the Dirichlet deformation method in the remainder of this paper. Suppose $$ V\in L^1_{\text{\rom{loc}}} (\Bbb R) \text{ is real valued}, \tag 2.1 $$ introduce the differential expression $\tau=-\frac{d^2}{dx^2}+V(x)$, $x\in\Bbb R$, and pick $\lambda_0 \in\Bbb R$ and $\eta_\pm (\lambda _0, x)$ satisfying $$\gathered \tau\psi(\lambda _0)=\lambda_0 \psi (\lambda _0), \\ \eta_\pm (\lambda _0, \,\cdot\,) \in L^2 ((R, \pm\infty)), \quad R\in\Bbb R, \ \eta_\pm (\lambda _0, x) {\text{ real-valued}}. \endgathered \tag 2.2 $$ Given $\eta_\pm (\lambda _0, x)$ we define the self-adjoint base (background) operator $H$ in $L^2 (\Bbb R)$ via $$\aligned Hf &=\tau f, \\ f\in\Cal D(H) &= \{g\in L^2 (\Bbb R) \mid g,g'\in AC_{\text{\rom{loc}}}(\Bbb R); \tau g\in L^2(\Bbb R); \\ &\qquad \lim\limits_{x\to\pm\infty} W(\eta_\pm (\lambda _0),g)(x) =0 \text{ if $\tau$ is l.c.~at $\pm\infty$}\}. \endaligned \tag 2.3 $$ Here $W(f,g)(x)=f(x)g'(x)-f'(x)g(x)$ denotes the Wronskian of $f,g\in AC_{\text{\rom{loc}}} (\Bbb R)$ (the set of locally absolutely continuous functions on $\Bbb R$) and l.c.~and l.p.~abbreviate the limit point and limit circle cases, respectively. The corresponding boundary condition at $\omega\infty$ in (2.3) is superfluous and hence to be deleted whenever $\tau$ is l.p.~at $\omega\infty$, $\omega\in\{-, +\}$. The reader unwilling to get caught up in limit circle situations may safely add the assumption that $\tau$ is l.p.~at $\pm\infty$ which renders $H$ independent of the choice of $\eta_\pm (\lambda_0, x)$. However, as discussed in Lemma 5.3, assuming $\tau$ to be l.p.~at $\pm\infty$ does not necessarily dispose of all limit circle considerations in connection with the deformation method at hand. Given $H$ and a fixed reference point $x_0 \in\Bbb R$, we introduce the associated Dirichlet operator $H^D$ in $L^2 (\Bbb R)$ by $$\aligned H^D f &= \tau f, \\ f\in \Cal D(H^D) &= \{g\in L^2 (\Bbb R) \mid g\in AC_{\text{\rom{loc}}}(\Bbb R), g'\in AC_{\text{\rom{loc}}} (\Bbb R\backslash \{x_0\}); \lim\limits_{\epsilon\downarrow 0} g(x_0 \pm\epsilon)=0; \\ &\qquad \tau g\in L^2 (\Bbb R); \lim\limits_{x\to\pm\infty} W(\eta_\pm (\lambda_0), g) (x)=0 \text{ if $\tau$ is l.c.~at $\pm\infty$}\}. \endaligned \tag 2.4 $$ Clearly, $H^D$ decomposes into $$ H^D = H^D_- \oplus H^D_+ \tag 2.5 $$ with respect to the orthogonal decomposition $$ L^2 (\Bbb R) =L^2 ((-\infty, x_0)) \oplus L^2 ((x_0, \infty)). \tag 2.6 $$ (For notational convenience, we shall later identify $(x_0, \sigma\infty)$ with $(-\infty, x_0)$ or $(x_0, \infty)$ depending on whether $\sigma =-$ or $\sigma =+$.) Moreover, for any $\mu\in\sigma_d (H^D)\backslash \sigma(H)$ ($\sigma_d (\,\cdot\,)=\sigma(\,\cdot\,) \backslash\sigma_{\text{\rom{ess}}}(\,\cdot\,)$, the discrete spectrum, $\sigma(\,\cdot\,)$ and $\sigma_{\text{\rom{ess}}}(\,\cdot\,)$, the spectrum and essential spectrum, respectively), we introduce the Dirichlet datum $$ (\mu,\sigma) \in\{\sigma_d (H^D)\backslash\sigma_d (H)\} \times \{-,+\}, \tag 2.7 $$ which identifies $\mu$ as a discrete Dirichlet eigenvalue on the interval $(x_0, \sigma \infty)$, that is, $\mu\in\sigma_d (H^D_\sigma)$, $\sigma\in \{-,+\}$ (but excludes it from being simultaneously a Dirichlet eigenvalue on $(x_0, -\sigma\infty)$). In some cases, for instance, if $V(x)\to\infty$ as $|x|\to\infty$, the spectrum and Dirichlet data uniquely determine $V(x)$ [\gstrans], Theorem~3.6 (cf.~also Remarks 4.9 and 4.10). Next, we pick a fixed spectral gap $(E_0, E_1)$ of $H$, the endpoints of which (without loss of generality) belong to the spectrum of $H$, $$ (E_0, E_1) \subset \Bbb R\backslash\sigma(H), \quad E_0, E_1 \in\sigma(H) \tag 2.8 $$ and choose a discrete eigenvalue $\mu$ of $H^D$ in the closure of that spectral gap, $$ \mu\in\sigma_d (H^D)\cap [E_0, E_1] \tag 2.9 $$ (we note there is at most one such $\mu$ since $(H^D -z)^{-1}$ is a rank-one perturbation of $(H-z)^{-1}$). According to (2.7), this either gives rise to a Dirichlet datum $$ (\mu,\sigma)\in (E_0, E_1) \times \{-,+\}, \tag 2.10 $$ or else to a discrete eigenvalue of $H^D_-$ and $H^D_+$, that is, $$ \mu\in\{E_0, E_1\}\cap\sigma_d (H)\cap\sigma_d(H^D_-)\cap \sigma_d (H^D_+) \tag 2.11 $$ since the eigenfunction of $H$ associated with $\mu$ has a zero at $x_0$. In particular, since $(H^D -z)^{-1}$ is a rank-one perturbation of $(H-z)^{-1}$, one infers $$ \sigma_{\text{\rom{ess}}} (H^D) = \sigma_{\text{\rom{ess}}}(H), \tag 2.12 $$ and thus, $\mu\in \{E_0, E_1\}\cap\sigma_{\text{\rom{ess}}}(H)$ is excluded by hypothesis (2.9). Hence, the case distinctions (2.10) and (2.11) are exhaustive. In addition to $\mu$ as in (2.9)--(2.11), we also need to introduce $\tilde\mu\in [E_0, E_1]$ and $\tilde\sigma\in \{-,+\}$ as follows: Either $$ (\tilde\mu, \sigma)\in (E_0, E_1) \times \{-,+\}, \tag 2.13 $$ or else $$ \tilde\mu\in \{E_0, E_1\}\cap\sigma_d (H). \tag 2.14 $$ Given $H$, we define solutions $\psi_\pm (z,x)$ of $(\tau-z)\psi(z)=0$ which satisfy $$\gathered \psi_\pm (z,\,\cdot\,)\in L^2 ((R,\pm\infty)), \quad R\in \Bbb R, \\ \lim\limits_{x\to\pm\infty} W(\psi_\pm (z), g)(x)=0 \quad \text{for all $g\in\Cal D(H)$}. \endgathered \tag 2.15 $$ If $\psi_\pm (z,x)$ exist, they are unique up to constant multiples. In particular, $\psi_\pm (z,x)$ exist for $z\in\Bbb C\backslash\sigma_{\text{\rom{ess}}}(H)$ and we can (and will) assume them to be holomorphic with respect to $z\in\Bbb C\backslash\sigma(H)$ and real-valued for $z\in\Bbb R$. One can choose, $$ \psi_\pm (z,x) =((H-z)^{-1}\chi_{(a,b)})(x) \quad\text{for } x\smat{>b}{a}$. (Here $\chi_\Omega (\,\cdot\,)$ denotes the characteristic function of a set $\Omega\subseteq\Bbb R$.) A finite number of isolated eigenvalues can be included in the domain of holomorphy of $\psi_\pm (z)$ by multiplying (2.16) with an appropriate function of $z$. Next, we state a simple technical result which will be needed in the context of (2.19) and (2.20). \proclaim{Lemma 2.1} Let $\psi_\sigma (\mu,\,\cdot\,)$, $\psi_\sigma(\tilde\mu, \,\cdot\,)\in L^2 ((R, \sigma\infty))$, $R\in\Bbb R$ be defined as in {\rom{(2.15)}}. Then $$ \lim\limits_{\tilde{\mu}\to\mu} (\tilde\mu -\mu)^{-1} W(\psi_\sigma (\mu), \psi_\sigma(\tilde\mu))(x) =-\int\limits^x_{\sigma\infty} dx' \psi_\sigma (\mu, x')^2. \tag 2.17 $$ \endproclaim \demo{Proof} Since $$ (\tilde\mu -\mu)^{-1} W(\psi_\sigma (\mu), \psi_\sigma (\tilde\mu))(x)=- \int\limits^x_{\sigma\infty} dx'\psi_\sigma (\mu, x') \psi_\sigma (\tilde\mu, x'), \tag 2.18 $$ ((2.18) is easily verified by differentiating $W(\psi_\sigma (\mu), \psi_\sigma (\tilde\mu))(x)$ w.r.t.~$x$ and using (2.15)), we only need to justify interchanging the limit $\tilde\mu \to\mu$ and the integral in (2.18). By (2.16), $$ ((H-\tilde\mu)^{-1}\chi_{(a,b)}) (x') = c_\pm(\tilde\mu) \psi_\pm (\tilde\mu, x') \text{ for } \aligned x' &>x>b>a \\ x' &< x\mu$, and abbreviate $$ W(x)=W(\psi_- (\mu), \psi_- (\tilde\mu))(x), \quad x\in\Bbb R. $$ Suppose that $$ W(x_1) =0 \quad \text{for some $x_1 \in\Bbb R$}. \tag 2.24 $$ Define $$ \eta_1 (x) = \cases \psi_- (\mu, x), & x\leq x_1 \\ \gamma_1 \psi_+ (\tilde\mu, x), & x\geq x_1 \endcases , $$ where $\gamma_1 \in\Bbb R$ is defined such that $\eta_1 \in\Cal D(H)$ and $$ \tilde\eta_1 (x) = \cases \psi_- (\mu, x), & x\leq x_1 \\ -\gamma_1 \psi_+ (\tilde\mu, x), & x>x_1 \endcases . $$ If $\tilde\mu \in\sigma_d (H)$, we define in addition $$ \eta_0 (x)=\psi_+ (\tilde\mu, x) =-\tilde\eta_0 (x), \quad x_0 =-\infty $$ and if $\mu\in\sigma_d (H)$, $$ \eta_2 (x)=\psi_- (\mu, x) =\tilde\eta_2 (x), \quad x_2 =+\infty. $$ Then $$ (\eta_j, \eta_k) = (\tilde\eta_j, \tilde\eta_k) \quad \text{for all $j,k$}. \tag 2.25 $$ Indeed, let $j\mu$). Define $$ \widehat W(x)=W(\psi_- (\mu), \psi_- (\tilde\mu))(x), \quad x\in\Bbb R $$ and suppose $$ \widehat W (x_1)=0 \quad \text{for some $x_1 \in\Bbb R$}. \tag 2.27 $$ We introduce $$ \eta_1 (x) =\cases \psi_- (\mu,x)-\gamma_1 \psi_- (\tilde\mu, x), &x\leq x_1 \\ 0, & x\geq x_1 \endcases $$ (fixing $\gamma_1$ by demanding $\eta_1 \in\Cal D(H)$) and $$ \tilde\eta_1 (x) =\cases \psi_- (\mu, x) +\gamma_1 \psi_- (\tilde\mu, x), & x\leq x_1 \\ 0, & x>x_1 \endcases . $$ If $\tilde\mu \in\sigma_d(H)$, we introduce in addition $$ \eta_0 (x) =\psi_- (\tilde\mu, x)=-\tilde\eta_0 (x), \quad x_0 =+\infty $$ and if $\tilde\mu \in\sigma_d (H)$, $$ \eta_2 (x)=\psi_- (\mu,x)=\tilde\eta_2 (x), \quad x_0 =+\infty. $$ The rest of the proof is analogous to the case considered first: The $\eta_j$'s are linearly independent by considering their supports and $$ \int\limits^{x_1}_{-\infty} dx\,\psi_- (\mu, x) \psi_- (\tilde\mu, x) = (\mu -\tilde\mu)^{-1} \lim\limits_{c\downarrow -\infty} [\widehat W(x_1) -\widehat W(c)]=0 $$ since $\widehat W (x_1)=0$ by hypothesis, and both $\psi_- (\mu, x)$ and $\psi_- (\tilde\mu, x)$ satisfy the boundary conditions of $H$ at $-\infty$. Finally, the cases $\psi_+ (\mu, x)$, $\psi_\pm (\tilde\mu, x)$ can be obtained by reflection. \qed \enddemo Actually, Lemma 2.2 is only the tip of the iceberg. The principal results of [\gst] relate the number of zeros of appropriate Wronskians on an arbitrary interval $(a,b)$ of the type studied in this section to dimensions of spectral projections of general Sturm-Liouville operators on $(a,b)$. For a previous generalization of Sturm's separation theorem invoking the sign of Wronskians, see [\lei]. For later reference, we now summarize our basic assumptions on $V$, $\mu$, and $\tilde\mu$ in the following hypothesis. \example{(H.2.3)} (i) Suppose $V\in L^1_{\text{\rom{loc}}}(\Bbb R)$ to be real-valued. (ii) $$\aligned &(E_0, E_1)\subset\Bbb R\backslash\sigma(H), \quad E_0, E_1 \in\sigma(H), \\ &\mu\in\sigma_d (H^D), \quad (\mu,\sigma)\in (E_0, E_1)\times \{-,+\} \text{ or } \mu\in\{E_0, E_1\}\cap \sigma_d (H), \\ &(\tilde\mu,\tilde\sigma)\in (E_0, E_1)\times\{-,+\} \text{ or } \tilde\mu \in \{E_0, E_1\}\cap\sigma_d (H), \\ &\mu,\tilde\mu \in [E_0, E_1], \quad \mu\neq\tilde\mu \text{ or } (\tilde\mu, \tilde\sigma) = (\mu,-\sigma), \quad \mu\in (E_0, E_1). \endaligned $$ \endexample Next, we introduce various solutions of $(\tilde\tau_{(\tilde\mu,\tilde\sigma)}-z) \tilde\psi (z)=0$ needed in (2.32)--(2.35) to define the self-adjoint operator $\tilde H_{(\tilde\mu, \tilde\sigma)}$ in $L^2 (\Bbb R)$ associated with $\tilde\psi_{(\tilde\mu, \tilde\sigma)}$. Define $$\align \tilde\psi_{-\sigma}(\mu,x) &=\psi_{-\tilde\sigma}(\tilde\mu,x) \big/ W_{(\tilde\mu, \tilde\sigma)}(x), \tag 2.28 \\ \tilde\psi_{\tilde\sigma}(\tilde\mu, x) &= \psi_\sigma (\mu,x) \big/ W_{(\tilde\mu,\tilde\sigma)}(x), \quad \tilde\psi_{\tilde\sigma}(\tilde\mu, x_0)=0. \tag 2.29 \endalign $$ Then $$ (\tilde\tau_{(\tilde\mu,\tilde\sigma)}\tilde\psi_{-\sigma}(\mu))(x)=\mu \tilde\psi_{-\sigma}(\mu,x), \quad (\tilde\tau_{(\tilde\mu,\tilde\sigma)} \tilde\psi_{\tilde\sigma}(\tilde\mu))(x)=\tilde\mu \tilde\psi_{\tilde\sigma} (\tilde\mu, x) \tag 2.30 $$ and $$ \tilde\psi_{-\sigma}(\mu,x) \tilde\psi_{\tilde\sigma}(\tilde\mu, x) = [W_{(\tilde\mu,\tilde\sigma)}(x)^{-1}]'. \tag 2.31 $$ The Dirichlet deformation operator $\tilde H_{(\tilde\mu,\tilde\sigma)}$ associated with $\tilde\tau_{(\tilde\mu,\tilde\sigma)}$ in (2.20) is then defined as follows: $$\aligned \tilde H_{(\tilde\mu,\tilde\sigma)}f &=\tilde\tau_{(\tilde\mu,\tilde\sigma)} f, \\ f\in\Cal D(\tilde H_{(\tilde\mu,\tilde\sigma)}) &= \{g\in L^2(\Bbb R)\mid g,g'\in AC_{\text{\rom{loc}}}(\Bbb R); \tilde\tau_{(\tilde\mu, \tilde\sigma)} g\in L^2 (\Bbb R); \\ & \qquad g \text{ satisfies one of the b.c.'s in Cases I--III if } \\ & \qquad \tilde\tau_{(\tilde\mu,\tilde\sigma)} \text{ is l.c.~at $-\infty$ and/or $+\infty$}\}. \endaligned \tag 2.32 $$ The boundary conditions (b.c.'s) alluded to in (2.32) are chosen as follows: \medpagebreak \flushpar {\bf{Case I:}} Either $\tau$ is l.p.~at $\pm\infty$ or $\tilde\sigma =\sigma$. $$\aligned \lim\limits_{x\to\tilde\sigma \infty} W(\tilde\psi_{\tilde\sigma}(\tilde\mu), g)(x) &= 0 \text{ if $\tilde\tau_{(\tilde\mu,\tilde\sigma)}$ is l.c.~at $\tilde\sigma \infty$}, \\ \lim\limits_{x\to -\tilde\sigma\infty} W(\tilde\psi_{-\sigma}(\mu), g)(x) &=0 \text{ if $\tilde\tau_{(\tilde\mu,\tilde\sigma)}$ is l.c.~at $-\tilde\sigma\infty$}. \endaligned \tag 2.33 $$ \flushpar{\bf{Case II:}} $\tilde\sigma=-\sigma$, $\tau$ is l.c.~at $-\infty$ or $+\infty$, and $\mu\in\sigma_d (H)$. $$ \lim\limits_{x\to\omega\infty} W(\tilde\psi_{\tilde\sigma}(\tilde\mu), g)(x)=0 \text{ if $\tilde\tau_{(\tilde\mu,\tilde\sigma)}$ is l.c.~at $\omega\infty$}, \quad \omega\in\{-,+\}. \tag 2.34 $$ \flushpar{\bf{Case III:}} $\tilde\sigma=-\sigma$, $\tau$ is l.c.~at $-\infty$ or $+\infty$, and $\tilde\mu\in\sigma_d(H)$. $$ \lim\limits_{x\to\omega\infty} W(\tilde\psi_{-\sigma}(\mu), g)(x)=0 \text{ if $\tilde\tau_{(\tilde\mu,\tilde\sigma)}$ is l.c.~at $\omega\infty$}, \quad \omega\in\{-,+\}. \tag 2.35 $$ (Note that Case II $=$ Case III if $(\tilde\mu, \tilde\sigma) = (\mu, -\sigma)$.) As always, there is no boundary condition at $\omega\infty$ in (2.32) if $\tilde\tau_{(\tilde\mu,\tilde\sigma)}$ is l.p.~at $\omega\infty$, $\omega\in\{-,+\}$. Cases I--III, of course, are not exhaustive. We singled them out since they are the only situations where the spectra of $H$ and $\tilde H_{(\tilde\mu,\tilde\sigma)}$ are closely related (see (3.17) and the discussion at the end of Section 3). If $\tilde\mu\in\sigma_d(H)$, we will occasionally use the more appropriate notation $\tilde V_{\tilde\mu}(x)$, $\tilde\tau_{\tilde\mu}$, and $\tilde H_{\tilde\mu}$ (instead of $\tilde V_{(\tilde\mu,\tilde\sigma)}(x)$, $\tilde\tau_{(\tilde\mu,\tilde\sigma)}$, and $\tilde H_{(\tilde\mu,\tilde\sigma)}$, cf.~the comments following (2.21)). We conclude this section by introducing the Dirichlet operator $\tilde H^D_{(\tilde\mu,\tilde\sigma)}$ associated with $\tilde H_{(\tilde\mu,\tilde\sigma)}$ and the fixed reference point $x_0\in\Bbb R$, $$\aligned \tilde H^D_{(\tilde\mu,\tilde\sigma)} f &=\tilde\tau_{(\tilde\mu,\tilde\sigma)} f, \\ f\in\Cal D(\tilde H^D_{(\tilde\mu,\tilde\sigma)}) &= \bigl\{g\in L^2 (\Bbb R) \mid g\in AC_{\text{\rom{loc}}} (\Bbb R), g'\in AC_{\text{\rom{loc}}}(\Bbb R\backslash \{x_0\}); \lim\limits_{\epsilon\downarrow 0} g(x_0 \pm\epsilon)=0; \\ & \qquad \tilde\tau_{(\tilde\mu,\tilde\sigma)}g\in L^2 (\Bbb R); g \text{ satisfies one of the b.c.'s in Cases I--III if }\\ &\qquad \tilde\tau_{(\tilde\mu,\tilde\sigma)} \text{ is l.c.~at $-\infty$ and/or $+\infty$}\bigr\}. \endaligned \tag 2.36 $$ In analogy to (2.5), $\tilde H^D_{(\tilde\mu,\tilde\sigma)}$ decomposes into $$ \tilde H^D_{(\tilde\mu,\tilde\sigma)} = \tilde H^D_{(\tilde\mu,\tilde\sigma), -} \oplus \tilde H^D_{(\tilde\mu,\tilde\sigma), +} \tag 2.37 $$ with respect to (2.6). \vskip 0.3in \flushpar{\bf \S 3. Half-Line Weyl-Titchmarsh and Spectral Functions} \vskip 0.1in In this section we derive the Weyl-Titchmarsh $m$-functions for the Dirichlet deformation operator $\tilde H_{(\tilde\mu,\tilde\sigma)}$ and relate them to those of $H$. Moreover, we provide a complete spectral characterization of $\tilde H^D_{(\tilde\mu,\tilde\sigma), \pm}$ in terms of $H^D_\pm$. We start by introducing the transformation $$ U_{(\tilde\mu,\tilde\sigma)}(z): \cases AC_{\text{\rom{loc}}}(\Bbb R)\to L^1_{\text{\rom{loc}}}(\Bbb R) \\ f(x)\to\tilde f_{(\tilde\mu,\tilde\sigma)}(z,x) =f(x)-(z-\mu)^{-1} \tilde\psi_{-\sigma} (\mu, x)W(\psi_\sigma (\mu), f)(x), \, z\in\Bbb C\backslash \{\mu\} \endcases \tag 3.1 $$ and note that by inspection, $$ ((\tilde\tau_{(\tilde\mu,\tilde\sigma)}-z) U_{(\tilde\mu,\tilde\sigma)}(z)\psi(z))(x) =0 \text{ if and only if } ((\tau-z)\psi(z))(x)=0, \quad z\in\Bbb C\backslash \{\mu\}. \tag 3.2 $$ Moreover, one verifies $$\aligned \tilde f_{(\tilde\mu,\tilde\sigma)}(z,x) &= (U_{(\tilde\mu,\tilde\sigma)}(z)f)(x) \\ &= (z-\mu)^{-1} (z-\tilde\mu) f(x)-(z-\mu)^{-1} \psi_{\tilde\sigma} (\tilde\mu, x) W(\psi_{-\tilde\sigma} (\mu), f)(x), \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad f\in AC_{\text{\rom{loc}}}(\Bbb R), \ z\in\Bbb C \backslash \{\mu\}, \endaligned \tag 3.3 $$ $$\gathered (U_{(\tilde\mu,\tilde\sigma)} (\tilde\mu)\psi_{-\tilde\sigma}(\tilde\mu))(x)=0, \quad (U_{(\tilde\mu,\tilde\sigma)} (\tilde\mu)\psi_{\tilde\sigma}(\tilde\mu))(x)= (\tilde\mu -\mu)^{-1} W(\psi_{\tilde\sigma}(\tilde\mu), \psi_{-\tilde\sigma}(\tilde\mu)) \tilde\psi_{\tilde\sigma} (\tilde\mu, x), \\ \lim\limits_{z\to\mu}(z-\mu) (U_{(\tilde\mu,\tilde\sigma)}(z) \psi_\sigma (\mu))(x)=0, \\ \lim\limits_{z\to\mu} (z-\mu)(U_{(\tilde\mu,\tilde\sigma)}(z)\psi_{-\tilde\sigma} (\mu))(x)= -W(\psi_\sigma (\mu), \psi_{-\sigma}(\mu))\tilde\psi_{-\sigma}(\mu, x). \endgathered \tag 3.4 $$ In addition, if $$ (\tau -z)\psi(z)=0, \quad (\tau -\widehat z) \widehat\psi (\widehat z)=0, \quad z, \widehat z\in\Bbb C\backslash\{\mu\}, $$ then $$\gather W(\tilde\psi_{-\sigma}(\mu), U_{(\tilde\mu,\tilde\sigma)}(z)\psi(z))(x) = \frac {W(\psi_{-\tilde\sigma}(\tilde\mu), \psi (z))(x)} {W_{(\tilde\mu,\tilde\sigma)}(x)}, \tag 3.5 \\ W(\tilde\psi_{\tilde\sigma}(\tilde\mu), U_{(\tilde\mu,\tilde\sigma)}(z)\psi(z))(x) = \frac{z-\tilde\mu}{z-\mu} \, \frac{W(\psi_\sigma (\mu), \psi(z))(x)} {W_{(\tilde\mu,\tilde\sigma)}(x)}, \tag 3.6 \\ W(U_{(\tilde\mu,\tilde\sigma)}(z)\psi(z), U_{(\tilde\mu,\tilde\sigma)}(\widehat z) \widehat\psi (\widehat z))(x) = \frac{z-\tilde\mu}{z-\mu} \, W(\psi(z), \widehat\psi (\widehat z))(x) \\ \qquad \qquad + \ \frac{z-\widehat z}{(z-\mu)(\widehat z-\mu)} \, \frac{W(\psi_\sigma (\mu), \widehat\psi (\widehat z))(x) W(\psi_{-\tilde\sigma}(\tilde\mu), \psi(z))(x)}{W_{(\tilde\mu,\tilde\sigma)}(x)}. \tag 3.7 \endgather $$ Next, let $\phi(z,x),\theta(z,x)$ be the standard fundamental system of solutions of $(\tau -z)\psi(z)=0$, $z\in\Bbb C$ defined by $$ \phi(z,x_0)=\theta'(z,x_0)=0, \quad \phi'(z,x_0)=\theta(z,x_0)=1, \quad z\in\Bbb C \tag 3.8 $$ with $x_0 \in\Bbb R$ the reference point used in (2.4), and denote by $\tilde\theta_{(\tilde\mu,\tilde\sigma)}(z,x), \tilde\phi_{(\tilde\mu,\tilde\sigma)}(z,x)$ the analogous fundamental system of solutions of $(\tilde\tau_{(\tilde\mu,\tilde\sigma)}-z) \tilde\psi(z)=0$, $z\in\Bbb C$ satisfying (3.8). Since by definition (2.15), $\psi_\sigma (z,\,\cdot\,)\in L^2((R, \sigma\infty))$, $R\in\Bbb R$, $z\in\Bbb C\backslash \Bbb R$ satisfy the boundary conditions of $H$ near $\sigma\infty$ (if any) and, in particular, $\lim_{x\to\sigma\infty} W(\psi_\sigma (z),g)(x)=0$, $g\in\Cal D(H)$, one obtains $$ \psi'_\sigma (z,x_0)/\psi_\sigma (z,x_0) = m_\sigma (z), \quad \sigma\in\{-,+\}, \ z\in\Bbb C\backslash\Bbb R, \tag 3.9 $$ where $m_\sigma (z)$ denotes the Weyl-Titchmarsh $m$-function of $H$ with respect to the half-line $(x_0, \sigma\infty)$, $\sigma\in\{-,+\}$. Thus, $$\aligned (U_{(\tilde\mu,\tilde\sigma)}(z)f)(x_0) &= \frac{z-\tilde\mu}{z-\mu}\, f(x_0), \\ (U_{(\tilde\mu,\tilde\sigma)}(z) f)' (x_0) &= f'(x_0) -\frac{\tilde\mu -\mu}{z-\mu}\, m_{-\tilde\sigma}(\tilde\mu) f(x_0), \quad z,\tilde\mu\in\Bbb C\backslash\{\mu\}, \ f\in AC_{\text{\rom{loc}}}(\Bbb R), \endaligned \tag 3.10 $$ imply $$\aligned \tilde\phi_{(\tilde\mu,\tilde\sigma)}(z,x) &= (U_{(\tilde\mu,\tilde\sigma)}(z) \phi(z))(x), \\ \tilde\theta_{(\tilde\mu,\tilde\sigma)}(z,x) &= \frac{z-\mu}{z-\tilde\mu}\, (U_{(\tilde\mu,\tilde\sigma)}(z)\theta(z))(x) + \frac{\tilde\mu -\mu}{z-\tilde\mu}\, m_{-\tilde\sigma}(\tilde\mu) (U_{(\tilde\mu,\tilde\sigma)}(z)\phi(z))(x), \, z,\tilde\mu \in\Bbb C\backslash\{\mu\}. \endaligned \tag 3.11 $$ The case $(\tilde\mu,\tilde\sigma)=(\mu,-\sigma)$ in (3.10) and (3.11) can be obtained by a limiting procedure ($\pm m_\pm (z)$ being Herglotz, has at most one simple pole for $\lambda\in [E_0, E_1]$ with a negative residue), $$ \lim\limits_{\tilde\mu\to\mu}(\tilde\mu -\mu)m_\sigma (\tilde\mu) = \biggl(\,\int\limits^{x_0}_{\sigma\infty}dx \, \phi(\mu,x)^2\biggr)^{-1}, \quad \sigma\in\{-,+\} \tag 3.12 $$ (see, e.g., [\gstrans], Appendix A for a brief summary on Weyl $m$-functions). The following general fact on Weyl $m$-functions, which provides an effective tool for computing them in the context of $\tilde H_{(\tilde\mu,\tilde\sigma)}$, may well be of independent interest. \proclaim{Lemma 3.1} Let $\widehat V\in L^1_{\text{\rom{loc}}}(\Bbb R)$ be real-valued, $\widehat\tau =-\frac{d^2}{dx^2}+\widehat V(x)$, $x\in\Bbb R$, and $\widehat\eta_\sigma (\lambda, x)$, $\lambda\in\Bbb R$, $\sigma\in\{-,+\}$ non-zero real-valued solutions of $(\widehat\tau -\lambda)\widehat\psi(\lambda)=0$. Define the self-adjoint operator $\widehat H$ in $L^2 (\Bbb R)$ by $$\aligned \widehat Hf &=\widehat\tau f, \\ f\in\Cal D(\widehat H) &= \bigl\{g\in L^2 (\Bbb R) \mid g,g'\in AC_{\text{\rom{loc}}} (\Bbb R); \widehat\tau g\in L^2 (\Bbb R); \\ &\qquad \lim\limits_{x\to\pm\infty} W(\widehat\eta_\pm (\lambda), g)(x) =0 \text{ if $\widehat\tau$ is l.c.~at $\pm\infty$}\bigr\}. \endaligned \tag 3.13 $$ \rom(If $\widehat\tau$ is l.p.~at $+\infty$ and/or $-\infty$, the corresponding boundary condition in {\rom{(3.13)}} is to be deleted.\rom) Denote by $\widehat\phi (z,x), \widehat\theta (z,x)$ the fundamental system of solutions of $(\widehat\tau -z) \widehat\psi(z)=0$, $z\in\Bbb C$, with initial values as in {\rom{(3.8)}}. Then the limits $$ \widehat m_\sigma (z) =-\lim\limits_{x\to\sigma\infty} \frac{W(\widehat\eta_\sigma (\lambda), \widehat\theta (z))(x)}{W(\widehat\eta_\sigma (\lambda), \widehat\phi(z)) (x)}, \quad z\in\Bbb C\backslash\Bbb R, \ \sigma\in\{-,+\} \tag 3.14 $$ exist and represent the Weyl-Titchmarsh $m$-functions of $\widehat H$ on the half-line $(x_0, \sigma\infty)$. \endproclaim \demo{Proof} First suppose that $\widehat\tau$ is l.p.~at $\sigma\infty$ and $z\in \Bbb C\backslash\Bbb R$. Then $$ -\frac{W(\widehat\eta_\sigma (\lambda), \widehat\theta(z))(x)}{W(\widehat\eta_\sigma (\lambda), \widehat\phi(z))(x)} = -\frac{\widehat\theta(z,x) + \tan(\beta(x))\widehat\theta' (z,x)}{\widehat\phi (z,x) + \tan(\beta(x))\widehat\phi' (z,x)}, \quad \cot(\beta(x))= -\eta'_\sigma (\lambda, x)/\eta_\sigma (\lambda, x) \tag 3.15 $$ converges to $\widehat m_\sigma (z)$. This does not quite represent the typical Weyl limit point consideration in which one usually involves an $x$-independent boundary condition parameter $\beta\in [0,\pi)$. However, due to the l.p.~hypothesis made, the Weyl disks shrink to a limit point and the $x$-dependence of $\beta(x)$ in (3.15) becomes immaterial in the limit $x\to\sigma\infty$. Next, assume $\widehat\tau$ is l.c.~at $\sigma\infty$. Then $\widehat\phi (z,\,\cdot\,), \widehat\theta (z,\,\cdot\,)\in L^2 ((R, \sigma\infty))$, $R \in\Bbb R$, and hence the limits $$\multline \lim\limits_{x\to\sigma\infty} W(\widehat\eta_\sigma (\lambda), \widehat\chi(z))(x) = W(\widehat\eta_\sigma (\lambda), \widehat\chi(z))(x_0) \\ + (z-\lambda) \int\limits^{x_0}_{\sigma\infty} dx' \, \widehat\eta_\sigma (\lambda, x') \widehat\chi (z,x') \text{ for } \widehat\chi(z,x) = \cases \widehat\phi (z,x) \\ \widehat\theta (z,x) \endcases \endmultline \tag 3.16 $$ exist. Actually, the limits in (3.16) not only exist but they are also non-zero since otherwise one could construct self-adjoint operators with boundary conditions at $\sigma\infty$ induced by $\widehat\phi(z,x)$ or $\widehat\theta(z,x)$ with associated eigenvalue $z\in\Bbb C\backslash\Bbb R$. Next, consider the function $$ \widehat\psi_{\sigma}(z,x) =\widehat\theta (z,x)+\widehat m_{\sigma} (z) \widehat\phi (z,x), \quad z\in\Bbb C\backslash\Bbb R, $$ where $\widehat m_{\sigma}(z)$ denotes the $m$-function associated with $\widehat H$ on $(x_0, \sigma\infty)$. Since by construction, $\widehat\psi_{\sigma}(z,x)$ satisfies the boundary conditions of $\widehat H$ at $\sigma\infty$, one infers $$\align 0 &=\lim\limits_{x\to\sigma\infty} W(\widehat\eta_{\sigma} (\lambda), \widehat\psi_{\sigma} (z))(x) \\ &= \bigl[ \lim\limits_{x\to\sigma\infty} W(\widehat\eta_{\sigma}(\lambda), \widehat\theta (z))(x) \bigr] +\widehat m_{\sigma}(z) \bigl[\lim\limits_{x\to\sigma\infty} W(\widehat\eta_{\sigma}(\lambda), \widehat\phi(z))(x)\bigr], \endalign $$ and hence (3.14) again. \qed \enddemo Applying Lemma 3.1 to $\tilde H_{(\tilde\mu, \tilde\sigma)}$, we obtain as our first major result the following expression for the half-line Weyl-Titchmarsh $m$-functions of $\tilde H_{(\tilde\mu, \tilde\sigma)}$ in terms of those of $H$. \proclaim{Theorem 3.2} Assume {\rom{(H.2.3)}}. Given $H$ and $\tilde H_{(\tilde\mu, \tilde\sigma)}$ by {\rom{(2.3)}} and {\rom{(2.32)}}, respectively, denote by $m_\pm (z)$ and $\tilde m_{(\tilde\mu, \tilde\sigma), \pm}(z)$ the corresponding $m$-functions associated with the half-lines $(x_0, \pm\infty)$. Then $$\aligned \tilde m_{(\tilde\mu, \tilde\sigma), \pm}(z) &= \frac{z-\mu}{z-\tilde\mu}\, m_\pm (z) -\frac{\tilde\mu-\mu}{z-\tilde\mu}\, m_{-\tilde\sigma}(\tilde\mu), \quad \tilde\mu\neq\mu, \\ \tilde m_{(\tilde\mu, \tilde\sigma), \pm}(z) &= m_\pm (z) - \biggl(\,\int\limits^{x_0}_{\sigma \infty}dx\, \phi(\mu,x)^2\biggr)^{-1} \frac{1}{z-\mu}, \quad (\tilde\mu, \tilde\sigma) = (\mu, -\sigma), \ z\in\Bbb C\backslash\Bbb R. \endaligned \tag 3.17 $$ \endproclaim \demo{Proof} Combining (3.11), (3.5), and (3.6), one computes $\tilde m_{(\tilde\mu, \tilde\sigma),\omega}(z)$, $\omega\in\{-,+\}$ either from $$ -\lim\limits_{x\to\omega\infty}\frac{W(\tilde\psi_{\tilde\sigma}(\tilde\mu), \tilde\theta_{(\tilde\mu, \tilde\sigma)}(z))(x)}{W(\tilde\psi_{\tilde\sigma}(\tilde\mu), \tilde\phi_{(\tilde\mu, \tilde\sigma)}(z))(x)} = \frac{z-\mu}{z-\tilde\mu} \biggl[-\lim\limits_{x\to\omega\infty} \frac{W(\psi_\sigma (\mu), \theta(z))(x)} {W(\psi_\sigma (\mu), \phi(z))(x)}\biggr] -\frac{\tilde\mu -\mu}{z-\tilde\mu}\, m_{-\tilde\sigma}(\tilde\mu) \tag 3.18 $$ or from $$\split -\lim\limits_{x\to\omega\infty}& \frac{W(\tilde\psi_{-\sigma}(\mu), \tilde\theta_{(\tilde\mu, \tilde\sigma)}(z))(x)}{W(\tilde\psi_{-\sigma}(\mu), \tilde\phi_{(\tilde\mu, \tilde\sigma)}(z))(x)} \\ & = \frac{z-\mu}{z-\tilde\mu} \biggl[-\lim\limits_{x\to\omega\infty} \frac{W(\psi_{-\tilde\sigma} (\tilde\mu), \theta(z))(x)}{W(\psi_{-\tilde\sigma} (\tilde\mu), \phi(z))(x)}\biggr] - \frac{\tilde\mu -\mu}{z-\tilde\mu}\, m_{-\tilde\sigma}(\tilde\mu) \endsplit\tag 3.19 $$ (or from both), depending on Cases I--III in (2.32)--(2.35) by applying Lemma 3.1 to $\tilde H_{(\tilde\mu, \tilde\sigma)}$ and $H$. \qed \enddemo An examination of $\tilde m_{(\tilde\mu, \tilde\sigma), \pm}(z)$ in (3.17) then reveals the following behavior near $\mu$ and $\tilde\mu$. \proclaim{Corollary 3.3} {\rom{(i)}} Suppose $\mu,\tilde\mu\in (E_0, E_1)$, $\tilde\mu \neq\mu$ or $\mu\in\{E_0, E_1\}\cap\sigma_d (H)$, $\tilde\mu\in (E_0, E_1)$. Then, $\tilde m_{(\tilde\mu, \tilde\sigma), -\tilde\sigma}(z)$ is holomorphic in a neighborhood of $\tilde\mu$ whereas $\tilde m _{(\tilde\mu, \tilde\sigma), \tilde\sigma}(z)$ has a simple pole at $\tilde\mu$ with residue $$ \lim\limits_{z\to\tilde\mu} (z-\tilde\mu) \tilde m_{(\tilde\mu, \tilde\sigma), \tilde\sigma} (z)= (\tilde\mu -\mu) [m_{\tilde\sigma}(\tilde\mu)-m_{-\tilde\sigma}(\tilde\mu)] \neq 0. \tag 3.20 $$ Both $\tilde m_{(\tilde\mu, \tilde\sigma), \pm}(z)$ are holomorphic in a neighborhood of $\mu$. {\rom{(ii)}} Assume $\mu=\tilde\mu\in (E_0, E_1)$, $\tilde\sigma =-\sigma$. Then $\tilde m_{(\tilde\mu, \tilde\sigma), -\tilde\sigma}(z)$ is holomorphic in a neighborhood of $\tilde\mu$ whereas $\tilde m_{(\tilde\mu, \tilde\sigma), \tilde\sigma}(z)$ has a simple pole at $\tilde\mu$ with residue $$ \lim\limits_{z\to\tilde\mu} (z-\tilde\mu)\tilde m_{(\tilde\mu, \tilde\sigma), \tilde\sigma} (z)=-\lim\limits_{\tilde\mu\to\mu} (\tilde\mu-\mu)m_\sigma (\tilde\mu) = -\biggl( \, \int\limits^{x_0}_{\sigma\infty} dx\,\phi(\mu,x)^2 \biggr)^{-1}. \tag 3.21 $$ {\rom{(iii)}} Assume $\mu\in (E_0, E_1)$, $\tilde\mu\in \{E_0, E_1\} \cap \sigma_d(H)$ or $\mu,\tilde\mu\in \{E_0, E_1\} \cap \sigma_d(H)$, $\mu\neq\tilde\mu$. Then $\tilde m_{(\tilde\mu, \tilde\sigma), \pm}(z)$ are both holomorphic in a neighborhood of $\mu$ and $\tilde\mu$. \endproclaim \demo{Proof} Observing that $$ m_\sigma (z) \operatornamewithlimits{=}\limits_{z\to\mu} c_\sigma (z-\mu)^{-1} + O(1), \ c_\sigma\in\Bbb R\backslash\{0\}, \ m_{-\sigma}(z) \operatornamewithlimits{=}\limits_{z\to\mu} O(1), \quad \mu\in (\lambda_0, \lambda_1), \tag 3.22 $$ $$ m_\pm (z) \operatornamewithlimits{=}\limits_{z\to\mu} c_\pm (z-\mu)^{-1} +O(1), \ c_\pm \in\Bbb R\backslash\{0\}, \ c_\pm\lessgtr 0, \ \mu\in\sigma_d (H^D)\cap\sigma (H), \tag 3.23 $$ cases (i) and (ii) are a straightforward consequence of (3.12),(3.17), and the fact that $m_- (\tilde\mu)\neq m_+ (\tilde\mu)$ since $\tilde\mu\notin\sigma_d (H)$. For (iii) one observes, in addition, that first, $$ m_- (\tilde\mu)=m_+ (\tilde\mu) \quad\text{since }\tilde\mu\in\sigma_d (H) \backslash\sigma_d (H^D) $$ by hypothesis, and second, $$ \lim\limits_{z\to\lambda} (z-\lambda) m_\pm (z) \text{ exists for all $\lambda\in [E_0, E_1]$} \tag 3.24 $$ since $\pm m_\pm (z)$ are Herglotz functions (and $\sigma_{\text{\rom{ess}}} (H)\cap [E_0, E_1]=\emptyset$). \qed \enddemo As we will explore in more detail in the next section, Corollary 3.3(i) for $\mu,\tilde\mu \in(E_0, E_1)$, $\mu\neq\tilde\mu$ just means the Dirichlet datum $(\mu, \sigma)$ gets changed into $(\tilde\mu, \tilde\sigma)$ and (ii) illustrates the ``flip" of the Dirichlet eigenvalue $\mu$ from one half-line $(x_0, \sigma\infty)$ to the other, $(x_0, -\sigma\infty)$, changing $(\mu,\sigma)$ into $(\mu, -\sigma)$. The remaining cases represent non-isospectral deformations of $H$ where the eigenvalue $\mu\in\sigma_d (H)$, respectively, $\tilde\mu\in\sigma_d (H)$, or both, $\mu,\tilde\mu\in\sigma_d (H)$ are actually ``knocked out" of the spectrum of $H$ (i.e., do not belong to $\sigma (\tilde H_{(\tilde\mu, \tilde\sigma)})$, respectively, $\sigma(\tilde H_{\tilde\mu})$). \proclaim{Corollary 3.4} Let $z\in\Bbb C\backslash\{\sigma(H)\cup\{\mu\}\}$, $\tilde\mu\in (E_0, E_1)$. Then $\tilde\psi_{\tilde\sigma}(\tilde\mu, x)$, $\tilde\psi_{-\sigma}(\mu,x)$, $\mu\neq\tilde\mu$, and $(U_{(\tilde\mu, \tilde\sigma)} (z))\psi_\pm (z))(x)$ satisfy the boundary conditions of $\tilde H_{(\tilde\mu, \tilde\sigma)}$ \rom(if any\rom) at $\tilde\sigma\infty$, $-\sigma\infty$, and $\pm\infty$, respectively. In particular, $$\aligned \lim\limits_{x\to\tilde\sigma\infty}W(\tilde\psi_\sigma (\tilde\mu), \tilde g)(x)&=0, \\ \lim\limits_{x\to -\sigma\infty} W(\tilde\psi_{-\sigma}(\mu),\tilde g)(x)&=0, \quad \mu\neq\tilde\mu, \\ \lim\limits_{x\to\pm\infty}W(U_{(\tilde\mu, \tilde\sigma)}(z)\psi_\pm (z), \tilde g)(x) &=0 \endaligned \tag 3.25 $$ for all $\tilde g\in\Cal D(\tilde H_{(\tilde\mu, \tilde\sigma)})$. Moreover, $$\aligned \psi_{\tilde\sigma}(\tilde\mu, \,\cdot\,) &\in L^2 ((R,\tilde\sigma\infty)), \\ \tilde\psi_{-\sigma}(\mu,\,\cdot\,) &\in L^2 ((R, -\sigma\infty)), \quad \mu\neq\tilde\mu, \\ U_{(\tilde\mu, \tilde\sigma)}(z)\psi_\pm (z) &\in L^2 ((R,\pm\infty)), \quad R\in\Bbb R \endaligned \tag 3.26 $$ \rom(justifying the notation we chose for $\tilde\psi_{\tilde\sigma}(\tilde\mu, x)$ and $\tilde\psi_{-\sigma}(\mu, x)$\rom). \endproclaim \demo{Proof} Let $\tilde H^D_{(\tilde\mu, \tilde\sigma),\omega}$ denote the Dirichlet operators (2.37) corresponding to $\tilde\tau_{(\tilde\mu, \tilde\sigma)}$ on the half-line $(x_0, \omega\infty)$, $\omega\in\{-,+\}$. Since $\tilde m_{(\tilde\mu, \tilde\sigma), \tilde\sigma}(z)$ has a pole at $z=\tilde\mu$ by Corollary 3.3, one infers $\tilde\mu\in \sigma_d (\tilde H^D_{(\tilde\mu, \tilde\sigma),\tilde\sigma})$. Moreover, since $(\tilde\tau_{(\tilde\mu, \tilde\sigma)}-\tilde\mu)\tilde\psi_{\tilde\sigma}(\tilde\mu)=0$ and $\tilde\psi_{\tilde\sigma}(\tilde\mu, x_0)=0$ (cf.~(2.29)), $\tilde\psi_{\tilde\sigma} (\tilde\mu, x)$ is the corresponding eigenfunction of $\tilde H^D_{(\tilde\mu, \tilde\sigma), \tilde\sigma}$ and hence $\tilde\psi_{\tilde\sigma}(\tilde\mu)\in \Cal D (\tilde H^D_{(\tilde\mu, \tilde\sigma), \tilde\sigma})$ satisfies (3.25) and (3.26). In the case of $\tilde\psi_{-\sigma}(\mu, x)$, $\mu\neq\tilde\mu$, one verifies that $$ \tilde m_{(\tilde\mu, \tilde\sigma), -\sigma}(\mu) =m_{-\tilde\sigma}(\tilde\mu) = \psi'_{-\tilde\sigma} (\tilde\mu, x_0)/\psi_{-\tilde\sigma} (\tilde\mu, x_0) = \tilde\psi'_{-\sigma}(\mu, x_0) \big/ \tilde\psi_{-\sigma} (\mu, x_0) $$ and hence (3.25) and (3.26) are valid for $\tilde\psi_{-\sigma}(\mu)$. Finally, as a consequence of (2.15), one infers that $$ \psi_\pm (z,x)=c_\pm (z) [\theta(z,x)+m_\pm (z)\phi (z,x)], \quad z\in\Bbb C\backslash \sigma(H) \tag 3.27 $$ for some constants $c_\pm (z)$. Combining (3.11), (3.17), and applying $U_{(\tilde\mu, \tilde\sigma)}(z)$ to (3.27) results in $$\multline ((U_{(\tilde\mu, \tilde\sigma)}(z) \psi_\pm (z))(x)= \\ c_\pm (z) \frac{z-\tilde\mu}{z-\mu}\, [\tilde\theta_{(\tilde\mu, \tilde\sigma)}(z,x) + \tilde m_{(\tilde\mu, \tilde\sigma)}(z) \tilde\phi_{(\tilde\mu, \tilde\sigma)}(z,x)], \quad z\in\Bbb C\backslash\{\sigma(H)\cup\{\mu\}\}. \endmultline \tag 3.28 $$ Clearly, (3.28) proves that $U_{(\tilde\mu, \tilde\sigma)}(z)\psi_\pm (z)$ satisfy (3.25) and (3.26). \qed \enddemo Given the fundamental relation between $\tilde m_{(\tilde\mu, \tilde\sigma),\pm}(z)$ and $m_\pm (z)$ in Theorem 3.2, we can readily derive the ensuing relation between the corresponding spectral functions $\tilde\rho_{(\tilde\mu, \tilde\sigma),\pm}(\lambda)$ and $\rho_\pm (\lambda)$ associated with the half-line Dirichlet operators $\tilde H^D_{(\tilde\mu, \tilde\sigma),\pm}$ and $H^D_\pm$. The right-continuous non-decreasing functions $\rho_\pm (\lambda)$ and $\tilde\rho_{(\tilde\mu, \tilde\sigma),\pm} (\lambda)$ are defined for $\lambda\in\Bbb R$ by $$ \rho_\pm (\lambda) -\rho_\pm (\lambda') = \pm\lim\limits_{\delta\downarrow 0} \lim\limits_{\epsilon\downarrow 0} \pi^{-1} \int\limits^{\lambda +\delta}_{\lambda' + \delta} d\nu \text{ Im}[m_\pm (\nu+i\epsilon)] \tag 3.29 $$ and $$ \tilde\rho_{(\tilde\mu, \tilde\sigma),\pm}(\lambda) -\tilde\rho_{(\tilde\mu, \tilde\sigma),\pm} (\lambda') =\pm\lim\limits_{\delta\downarrow 0} \lim\limits_{\epsilon\downarrow 0} \pi^{-1} \int\limits^{\lambda +\delta}_{\lambda' +\delta} d\nu \text{ Im} [m_{(\tilde\mu, \tilde\sigma), \pm}(\nu+ i\epsilon)]. \tag 3.30 $$ This sets up the second major result of this section. \proclaim{Theorem 3.5} Assume {\rom{(H.2.3)}}. Let $H^D_\pm$ and $\tilde H^D_{(\tilde\mu, \tilde\sigma), \pm}$ denote the Dirichlet operators {\rom{(2.4)}} and {\rom{(2.37)}}, and $d\rho_\pm (\lambda)$ and $d \tilde\rho_{(\tilde\mu, \tilde\sigma),\pm}(\lambda)$ the corresponding spectral measures generated by $\rho_\pm (\lambda)$ and $\tilde\rho_{(\tilde\mu, \tilde\sigma),\pm}(\lambda)$, respectively. Then $$\align d\tilde\rho_{(\tilde\mu, \tilde\sigma),\pm}(\lambda) &= \frac{\lambda-\mu} {\lambda-\tilde\mu}\, d\rho_\pm (\lambda) + (\tilde\mu -\mu) \left\{ \matrix \format\l &\quad\l \\ 0, & \tilde\sigma =\mp \\ [m_- (\tilde\mu)-m_+ (\tilde\mu)], & \tilde\sigma=\pm \endmatrix \right\} d\theta (\lambda-\tilde\mu), \ \mu\neq\tilde\mu, \\ d\tilde\rho_{(\tilde\mu, \tilde\sigma), \pm}(\lambda) &= d\rho_\pm (\lambda) + \left\{ \matrix \format\l &\quad\l \\ 0, & \sigma=\pm \\ (\pm\int^{x_0}_{\mp\infty} dx\, \phi(\mu, x)^2)^{-1}, & \sigma=\mp \endmatrix \right \} d\theta (\lambda-\mu), \tag 3.31 \\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {(\tilde\mu, \tilde\sigma) = (\mu, -\sigma), \ \lambda\in\Bbb R.} \endalign $$ Here $\theta(x)=\cases 1, & x\geq 0 \\ 0, & x<0 \endcases$. \endproclaim \demo{Proof} Inserting (3.17), for $\mu\neq\tilde\mu$ (for simplicity) into (3.30) yields $$\multline \tilde\rho_{(\tilde\mu, \tilde\sigma),\pm}(\lambda) -\tilde\rho_{(\tilde\mu, \tilde\sigma) \pm} (\lambda') = \pm \lim\limits_{\delta\downarrow 0} \lim\limits_{\epsilon\downarrow 0} \pi^{-1} \int\limits^{\lambda+\delta}_{\lambda' +\delta} d\nu \biggl\{ \frac{\epsilon^2 + (\nu-\tilde\mu)(\nu-\mu)}{(\nu-\tilde\mu)^2 +\epsilon^2}\, \text{Im}[m_\pm (\nu+i\epsilon)] \\ + \frac{(\tilde\mu -\mu)\epsilon}{(\nu-\tilde\mu)^2 +\epsilon^2} \{m_{-\tilde\sigma} (\tilde\mu) -\text{Re}[m_\pm (\nu+i\epsilon)]\}\biggr\}. \endmultline \tag 3.32 $$ Since $$\gather m_\pm (\nu) \text{ are real-valued for } \nu\in [E_0, E_1]\backslash \{\mu\}, \\ \text{Im}[m_\pm (\nu+ i0)]\,d\nu \text{ has no support in a sufficiently small} \\ \text{neighborhood of $\tilde\mu$ (since $\tilde\mu \in [E_0, E_1]\backslash\{\mu\}$)}, \tag 3.33\\ \pi^{-1} \frac{\epsilon\, d\nu}{(\nu-\tilde\mu)^2 +\epsilon^2} \underset \epsilon\downarrow 0 \to\longrightarrow d\theta(\nu-\tilde\mu) \quad\text{weakly}, \tag 3.34 \\ \pi^{-1} \text{Im}[m_\pm (\nu+i\epsilon)]\, d\nu \underset \epsilon\downarrow 0 \to\longrightarrow d\rho_\pm (\nu) \quad\text {weakly [49]}, \tag 3.35 \endgather $$ $$\gathered \lim\limits_{\epsilon\downarrow 0} (\mp i\epsilon\pi^{-1} m_\pm (\lambda +i\epsilon)) =\rho_\pm (\lambda) - \rho_\pm (\lambda -0), \\ \lim\limits_{\epsilon\downarrow 0} (\mp i\epsilon\pi^{-1} m_\pm (\lambda +i\epsilon)) \leq \rho_\pm (E_1)- \rho_\pm (E_0 -0), \quad \lambda\in [E_0, E_1], \\ |\epsilon\pi^{-1} m_\pm (\lambda + i\epsilon)| \leq C, \quad\epsilon\in [0, \epsilon_0], \ \lambda \in [E_0, E_1]\text{ for some } \epsilon_0 >0, \ C>0, \endgathered \tag 3.36 $$ (3.32) implies (3.31) for $\mu\neq\tilde\mu$ by splitting the integral in (3.32) into a sufficiently small interval around $\tilde\mu$ (if $\tilde\mu\in [\lambda', \lambda]$) and the remaining intervals (applying the dominated convergence theorem). The case $(\tilde\mu, \tilde\sigma)=(\mu, -\sigma)$ is treated analogously. \qed \enddemo \remark{Remark 3.6} If $\tilde\mu\neq\mu$, the factor $(\lambda-\mu)/(\lambda-\tilde\mu) \neq 1$ in (3.31) shows that the half-line Dirichlet deformation method $H^D_\pm \to \tilde H^D_{(\tilde\mu, \tilde\sigma), \pm}$ affects all remaining norming constants corresponding to eigenvalues of $H^D_\pm$. More precisely, denote by $$ c^2_{\pm, n} = \biggl( \mp \int\limits^{x_0}_{\pm\infty} dx\, \phi(\lambda_{\pm, n}, x)^2\biggr)^{-1} =\rho_\pm (\lambda_{\pm, n}) -\rho_\pm (\lambda_{\pm, n}-0) $$ the norming constant associated with $\lambda_{\pm, n}\in\sigma_p (H^D_\pm)$, $\lambda_{\pm, n}\neq\mu$ and denote by $\tilde c^2_{(\tilde\mu, \tilde\sigma), \pm, n}$ the one associated with $\lambda_{\pm, n}\in\sigma_p (\tilde H^D_{(\tilde\mu, \tilde\sigma), \pm})$. Then $$ \tilde c^2_{(\tilde\mu, \tilde\sigma), \pm, n} =\frac{\lambda_{\pm, n}-\mu} {\lambda_{\pm, n}-\tilde\mu}\, c^2_{\pm, n}. \tag 3.37 $$ Only in the case $(\tilde\mu, \tilde\sigma)=(\mu, -\sigma)$, the remaining norming constants stay invariant, $$ \tilde c^2_{(\mu, -\sigma), \pm, n}=c^2_{\pm, n}. \tag 3.38 $$ In fact, the deformation $(\mu, \sigma)\to (\mu, -\sigma)$ coincides with the isospectral case of the double commutation method considered in Appendix B (cf.~Remark B.3(i)). The corresponding invariance in (3.38) was originally proven in [\gjfa]. \endremark Theorem 3.5 implies the following half-line deformation result. \proclaim{Theorem 3.7} Assume {\rom{(H.2.3)}} and denote by $H^D_\pm$ and $\tilde H^D_{(\tilde\mu, \tilde\sigma), \pm}$ the half-line Dirichlet operators {\rom{(2.4)}} and {\rom{(2.37)}}. \roster \item"\rom{(i)}" Suppose $\mu, \tilde\mu \in (E_0, E_1)$. Then $$\aligned \sigma_{(p)} (\tilde H^D_{(\tilde\mu, \tilde\sigma), \tilde\sigma}) &= \cases \{\sigma_{(p)} (H^D_\sigma )\backslash \{\mu\}\}\cup \{\tilde\mu\}, &\sigma =\tilde\sigma \\ \sigma_{(p)} (H^D_{-\sigma})\cup \{\tilde\mu\}, & \sigma=-\tilde\sigma \endcases , \\ \sigma_{(p)} (\tilde H^D_{(\tilde\mu, \tilde\sigma), -\tilde\sigma}) & = \cases \sigma_{(p)}(H^D_\sigma)\backslash \{\mu\}, & \sigma=-\tilde\sigma \\ \sigma_{(p)}(H^D_{-\sigma}), &\sigma=\tilde\sigma \endcases , \quad \tilde\mu\neq\sigma (H^D_{(\tilde\mu, \tilde\sigma), -\tilde\sigma}). \endaligned \tag 3.39 $$ \item"\rom{(ii)}" Assume $\mu\in\{E_0, E_1\}\cap\sigma_d(H)$, $\tilde\mu\in (E_0, E_1)$. Then $$\aligned \sigma_{(p)}(\tilde H^D_{(\tilde\mu,\tilde\sigma), \tilde\sigma}) &= \{\sigma_{(p)} (H^D_{\tilde\sigma})\backslash \{\mu\}\}\cup\{\tilde\mu\}, \\ \sigma_{(p)} (H^D_{(\tilde\mu,\tilde\sigma), -\tilde\sigma}) &=\sigma_{(p)} (H^D_{-\tilde\sigma})\backslash\{\mu\}, \quad \tilde\mu\notin\sigma (H^D_{(\tilde\mu,\tilde\sigma), -\tilde\sigma}). \endaligned \tag 3.40 $$ \item"\rom{(iii)}" Suppose $\mu\in (E_0, E_1)$, $\tilde\mu\in \{E _0, E_1\} \cap\sigma_d (H)$. Then $$\aligned \sigma_{(p)}(\tilde H^D_{\tilde\mu,\sigma}) &= \sigma_{(p)}(H^D_\sigma)\backslash \{\mu\}, \\ \sigma_{(p)}(H^D_{\tilde\mu,-\sigma}) &=\sigma_{(p)}(H^D_{-\sigma}), \quad \tilde\mu\notin\sigma (H^D_{\tilde\mu,\pm}). \endaligned \tag 3.41 $$ \item"\rom{(iv)}" Assume $\mu,\tilde\mu\in \{E_0, E_1\}\cap\sigma_d (H)$, $\mu\neq\tilde\mu$. Then $$ \sigma_{(p)}(\tilde H^D_{\tilde\mu, \pm}) =\sigma_{(p)} (H^D_\pm)\backslash \{\mu\}, \quad \tilde\mu\notin\sigma (\tilde H^D_{\tilde\mu, \pm}). \tag 3.42 $$ \rom(Here $\sigma_{(p)}(\,\cdot\,)$ denotes $\sigma (\,\cdot\,)$ or $\sigma_p (\,\cdot\,)$ \rom(the point spectrum, i.e., the set of eigenvalues\rom) and we recall our occasional use of the notation of $\tilde H^D_{\tilde\mu,\pm}$ instead of $\tilde H^D_{(\tilde\mu, \tilde\sigma), \pm}$ if $\tilde\mu\in\sigma_d (H)$, cf.~the paragraph preceding Lemma 2.2.\rom) \item"\rom{(v)}" $$ \sigma_{\text{\rom{ess, ac, sc}}} (\tilde H^D_{(\tilde\mu,\tilde\sigma), \pm}) =\sigma_{\text{\rom{ess, ac, sc}}} (H^D_\pm). \tag 3.43 $$ Moreover, $\tilde H^D_{(\tilde\mu,\tilde\sigma),\pm}$ and $H^D_\pm$, restricted to the orthogonal complements of the \rom(at most one-dimensional, possibly equaling $\{0\}$\rom) eigenspaces corresponding to $\tilde\mu$ and $\mu$, are unitarily equivalent. \endroster \endproclaim \demo{Proof} This is a direct consequence of Corollary 3.3, Theorem 3.5, and the fact that half-line spectra corresponding to separated boundary conditions are simple. In particular, we note that by Corollary 3.4(i) and (iii), $\tilde m_{(\tilde\mu,\tilde\sigma), \pm} (z)$ are holomorphic in a sufficiently small neighborhood of $\mu$ and/or $\tilde\mu$ whenever they belong to $\sigma_d (H)$. \qed \enddemo As long as $\mu,\tilde\mu\in (E_0, E_1)$ and hence $\mu,\tilde\mu\notin \sigma_d(H)$, (3.39) just says that the Dirichlet datum $(\mu,\sigma)$ associated with $H^D=H^D_- \oplus H^D_+$ got changed into the Dirichlet datum $(\tilde\mu,\tilde\sigma)$ associated with $\tilde H^D_{(\tilde\mu,\tilde\sigma)}=\tilde H^D_{(\tilde\mu, \tilde\sigma), -} \oplus \tilde H^D_{(\tilde\mu,\tilde\sigma), +}$. The cases (ii)--(iv) examine all remaining possibilities where $\mu$ and/or $\tilde\mu$ belong to $\sigma_d (H)$ and possibly $\sigma_d (H^D_\pm)$ in which case, however, they no longer belong to $\sigma_d (\tilde H^D_{(\tilde\mu,\tilde\sigma)})$. We have yet to show that our choices I--III of boundary conditions of $\tilde H_{(\tilde\mu, \tilde\sigma)}$ in (2.32)--(2.35) are indeed the only ones that lead to our fundamental formula (3.17) as claimed after (2.35). We only need to focus on l.c.~cases and hence assume that $\tilde\tau$ is l.c.~at $\pm\infty$. By Lemma 3.1, the $m$-functions $\tilde m_{(\tilde\mu,\tilde\sigma), \pm}(z)$ of $\tilde H_{(\tilde\mu,\tilde\sigma)}$ can be computed as follows, $$ \tilde m_{(\tilde\mu,\tilde\sigma), \pm}(z) =-\lim\limits_{x\to\pm\infty} \frac {W(\tilde f(\lambda), \tilde\theta_{(\tilde\mu,\tilde\sigma)}(z))(x)} {W(\tilde f(\lambda), \tilde\phi_{(\tilde\mu,\tilde\sigma)}(z))(x)}, \tag 3.44 $$ where $(\tilde\tau_{(\tilde\mu,\tilde\sigma)}-\lambda)\tilde f(\lambda)=0$ for some $\lambda\in\Bbb R$. Consider a corresponding $f(\lambda, x)$ satisfying $(\tau - \lambda) f(\lambda)=0$ and $\tilde f(\lambda)=U_{(\tilde\mu,\tilde\sigma)}(\lambda) f(\lambda)$. Then (3.44) becomes $$ \tilde m_{(\tilde\mu,\tilde\sigma), \pm}(z) =\frac{z-\mu}{z-\tilde\mu} \biggl[ -\lim\limits_{x\to\pm\infty} \frac{W(U_{(\tilde\mu,\tilde\sigma)}(\lambda) f(\lambda), U_{(\tilde\mu,\tilde\sigma)}(z) \theta(z))(x)}{W(U_{(\tilde\mu,\tilde\sigma)}(\lambda) f(\lambda), U_{(\tilde\mu,\tilde\sigma)}(z)\phi(z))(x)}\biggr] - \frac{\tilde\mu -\mu} {z-\tilde\mu}\, m_{-\tilde\sigma}(\tilde\mu). \tag 3.45 $$ Applying (3.7) to (3.45) then yields $$\multline \frac{z-\tilde\mu}{z-\mu} \biggl\{ \tilde m_{(\tilde\mu,\tilde\sigma), \pm}(z) + \frac{\tilde\mu -\mu}{z-\tilde\mu}\, m_{-\tilde\sigma}(\tilde\mu)\biggr\} = -\lim\limits_{x\to\pm\infty} \biggl\{(\lambda-\tilde\mu) W(f(\lambda), \phi(z))(x) \\ - \frac{(\lambda-z)(\tilde\mu -\mu)}{z-\mu} \frac{W(\psi_\sigma (\mu), \phi(z))(x) W(f(\lambda), \psi_{-\sigma}(\tilde\mu))(x)}{W(\psi_\sigma(\mu), \psi_{-\tilde\sigma} (\tilde\mu))(x)}\biggr\} \biggl\{(\lambda-\tilde\mu) W(f(\lambda), \theta (z))(x) \\ -\frac{(\lambda-z) (\tilde\mu-\mu)}{z-\mu} \frac{W(\psi_\sigma (\mu),\theta(z))(x) W(f(\lambda), \psi_{-\sigma}(\tilde\mu))(x)}{W(\psi_\sigma (\mu), \psi_{-\tilde\sigma} (\tilde\mu))(x)}\biggr\}. \endmultline \tag 3.46 $$ In order to reproduce (3.17), the right-hand side of (3.46) would have to equal $$ -\lim\limits_{x\to\pm\infty} \frac{W(\eta(\lambda), \theta(z))(x)}{W(\eta(\lambda), \phi(z))(x)} =m_\pm (z) \tag 3.47 $$ for some real-valued solution $\eta(z,x)$ of $(\tau-\lambda)\psi(\lambda)=0$ which satisfies the boundary conditions of $H$ at $\pm\infty$. Clearly, $f(\lambda, x) = \psi_\sigma (\mu,x)$ and $f(\lambda, x)=\psi_{-\tilde\sigma} (\tilde\mu, x)$ are distinguished in (3.46) and these were precisely the cases we singled out in (2.32)--(2.35). No other choice of $f(\lambda, x)$ in (3.46) is compatible with (3.47). \vskip 0.3in \flushpar{\bf \S 4. Spectral and Weyl-Titchmarsh Matrices, Isospectral Deformations} \vskip 0.1in In this section we prove our principal results including explicit computations of the Weyl-Titchmarsh and spectral matrices of $\tilde H_{(\tilde\mu, \tilde\sigma)}$ in terms of those of $H$. Moreover, we provide a complete spectral characterization of $\tilde H_{(\tilde\mu, \tilde\sigma)}$ and $\tilde H^D_{(\tilde\mu, \tilde\sigma)}$ in terms of $H$ and $H^D$. We start with the Weyl-Titchmarsh matrices for $H$ and $\tilde H_{(\tilde\mu, \tilde\sigma)}$. To fix notation, we introduce the Weyl-Titchmarsh $M$-matrix in $\Bbb C^2$ associated with $H$ by $$\align M(z) &= (M_{p, q}(z))_{1\leq p, q\leq 2} \\ &= [m_- (z) -m_+ (z)]^{-1} \pmatrix m_- (z) m_+(z) & [m_- (z)+m_+ (z)]/2 \\ [m_- (z)+m_+ (z)]/2 & 1 \endpmatrix , \quad z\in\Bbb C\backslash\Bbb R \tag 4.1 \endalign $$ and similarly, in connection with $\tilde H_{(\tilde\mu, \tilde\sigma)}$, by $$\align \tilde M_{(\tilde\mu, \tilde\sigma)}(z) &= (\tilde M_{(\tilde\mu, \tilde\sigma), p,q} (z))_{1\leq p, q\leq 2} \\ &= [\tilde m_{(\tilde\mu, \tilde\sigma), -}(z) -\tilde m_{(\tilde\mu, \tilde\sigma), +}(z)]^{-1} \\ & \ \times \pmatrix \tilde m_{(\tilde\mu, \tilde\sigma), -}(z) \tilde m_{(\tilde\mu, \tilde\sigma), +}(z) & [\tilde m_{(\tilde\mu, \tilde\sigma), -}(z) + \tilde m_{(\tilde\mu, \tilde\sigma), +}(z)]/2 \\ [\tilde m_{(\tilde\mu, \tilde\sigma), -}(z) + \tilde m_{(\tilde\mu, \tilde\sigma), +}(z)]/2 & 1 \endpmatrix, \ z\in\Bbb C\backslash\Bbb R. \tag 4.2 \endalign $$ An application of Theorem 3.2 then yields \proclaim{Theorem 4.1} Assume {\rom{(H.2.3)}} and $z\in\Bbb C\backslash\Bbb R$. Given $H$ and $\tilde H_{(\tilde\mu, \tilde\sigma)}$ by {\rom{(2.3)}} and {\rom{(2.32)}}, respectively, their Weyl-Titchmarsh $M$-matrices are related by $$\align \tilde M_{(\tilde\mu, \tilde\sigma), 1,1}(z) &= \frac{z-\mu}{z-\tilde\mu}\, M_{1,1}(z) - 2\frac{\tilde\mu-\mu}{z-\tilde\mu}\, m_{-\tilde\sigma}(\tilde\mu)M_{1,2}(z) \\ &\qquad + \frac {(\tilde\mu-\mu)^2}{(z-\mu)(z-\tilde\mu)}\, m_{-\tilde\sigma} (\tilde\mu)^2 M_{2,2}(z), \tag 4.3 \\ \tilde M_{(\tilde\mu, \tilde\sigma), 1,2}(z) &= M_{1,2}(z) -\frac{\tilde\mu-\mu}{z-\mu}\, m_{-\tilde\sigma}(\tilde\mu) M_{2,2}(z), \tag 4.4 \\ \tilde M_{(\tilde\mu, \tilde\sigma),2,2}(z) &= \frac{z-\tilde\mu}{z-\mu}\, M_{2,2} (z), \quad \tilde\mu\neq\mu. \tag 4.5 \endalign $$ Equivalently, $$\split \tilde M_{(\tilde\mu, \tilde\sigma)}(z) &=(z-\mu)^{-1} (z-\tilde\mu)^{-1} \pmatrix z-\mu & -(\tilde\mu-\mu)m_{-\tilde\sigma}(\tilde\mu) \\ 0 & z-\tilde\mu \endpmatrix \\ &\qquad \times M(z) \pmatrix z-\mu & 0 \\ -(\tilde\mu -\mu)m_{-\tilde\sigma}(\tilde\mu) & z-\tilde\mu \endpmatrix , \quad z\in\Bbb C\backslash\Bbb R. \endsplit \tag 4.6 $$ \endproclaim The case $(\tilde\mu, \tilde\sigma)=(\mu, -\sigma)$ follows by a straightforward limiting argument (see (3.12), (3.17), and (3.31)). \demo{Proof} This is just a combination of (3.17), (4.1), and (4.2). \qed \enddemo We note that (4.3) can be written as $$ \tilde M_{(\tilde\mu, \tilde\sigma), 1,1}(z)=\frac{z-\mu}{z-\tilde\mu}\, M_{2,2}(z) \biggl[m_+ (z) -\frac{\tilde\mu-\mu}{z-\mu}\, m_{-\tilde\sigma}(\tilde\mu)\biggr] \biggl[m_- (z) -\frac{\tilde\mu -\mu}{z-\mu}\, m_{-\tilde\sigma}(\tilde\mu) \biggr]. \tag 4.7 $$ A close examination of $\tilde M_{(\tilde\mu, \tilde\sigma)}(z)$ then reveals the following behavior near $\mu$ and $\tilde\mu$. \proclaim{Corollary 4.2} $\tilde M_{(\tilde\mu, \tilde\sigma)}(z)$ is holomorphic in a neighborhood of $\mu$ and $\tilde\mu$ \rom(for all values of $\mu$ and $\tilde\mu$ admitted by {\rom{(H.2.3)}}\rom). \endproclaim \demo{Proof} It suffices to examine the pole structure (or better, the lack thereof) of $\tilde M_{(\tilde\mu, \tilde\sigma),p,p}(z)$, $p=1,2$ since $\det[\tilde M(z)]=-\frac{1} {4}$ then controls that one of $\tilde M_{(\tilde\mu, \tilde\sigma),1,2}(z)$ as well. The proof then proceeds along a case-by-case study depending on whether $\mu$, respectively $\tilde\mu$, lie in $(E_0, E_1)$ or in $\{E_0, E_1\}\cap \sigma_d (H)$. More specifically, one uses (3.26)--(3.28), $$\aligned &m_- (\tilde\mu) =m_+ (\tilde\mu), \ M_{2,2}(z) {\underset {z\to\tilde\mu} \to =} c(z-\tilde\mu)^{-1}+O(1), \quad c\in\Bbb R\backslash \{0\} \\ & \text{ if and only if }\tilde\mu\in \sigma_d(H)\backslash \sigma_d (H^D) \endaligned \tag 4.8 $$ and $$ M_{2,2}(z) \underset {z\to\mu} \to = c(z-\mu) +O((z-\mu)^2), \quad c\in\Bbb R \backslash \{0\}\text{ for } \mu\in [E_0, E_1]\cap\sigma_d (H^D). \tag 4.9 $$ The holomorphy assertion then follows directly from (4.5) and (4.7). \qed \enddemo Given the basic connection between $\tilde M_{(\tilde\mu, \tilde\sigma)}(z)$ and $M(z)$ in Theorem 4.1, we can now proceed to derive the analogous relations between the spectral matrices $\tilde\rho_{(\tilde\mu, \tilde\sigma)}(\lambda)$ and $\rho(\lambda)$ associated with $\tilde H_{(\tilde\mu, \tilde\sigma)}$ and $H$, respectively. The right-continuous non-decreasing matrices $\rho(\lambda)$ and $\tilde \rho_{(\tilde\mu, \tilde\sigma)}(\lambda)$ in $\Bbb C^2$ are defined for $\lambda\in\Bbb R$ by $$\gather \rho(\lambda) = (\rho_{p,q}(\lambda))_{1\leq p, q\leq 2}, \quad \tilde\rho_{(\tilde\mu, \tilde\sigma)}(\lambda) =(\tilde\rho_{(\tilde\mu, \tilde\sigma),p,q}(\lambda))_{1\leq p, q\leq 2}, \\ \rho(\lambda)- \rho(\lambda') =\lim\limits_{\delta\downarrow 0} \lim\limits_{\epsilon\downarrow 0} \pi^{-1} \int\limits^{\lambda+\delta}_{\lambda'+\delta} d\nu \, \text{Im}[M(\nu+i\epsilon)], \tag 4.10 \\ \tilde\rho_{(\tilde\mu, \tilde\sigma)}(\lambda) - \tilde\rho_{(\tilde\mu, \tilde\sigma)}(\lambda') =\lim\limits_{\delta\downarrow 0} \lim\limits_{\epsilon\downarrow 0} \pi^{-1} \int\limits^{\lambda+\delta}_{\lambda'+\delta} d\nu\, \text{Im}[\tilde M_{(\tilde\nu, \tilde\sigma)}(\nu+i\epsilon)]. \tag 4.11 \endgather $$ The result for $\tilde\rho_{(\tilde\mu, \tilde\sigma)}(\lambda)$ in terms of that of $\rho(\lambda)$ then reads as follows. \proclaim{Theorem 4.3} Assume {\rom{(H.2.3)}}. Given $H$ and $\tilde H_{(\tilde\mu, \tilde\sigma)}$ by {\rom{(2.3)}} and {\rom{(2.32)}}, let $d\rho(\lambda)$ and $d\tilde\rho_{(\tilde\mu, \tilde\sigma)}(\lambda)$ be the corresponding $\Bbb C^2$-valued spectral measures generated by $\rho(\lambda)$ and $\tilde\rho_{(\tilde\mu, \tilde\sigma)} (\lambda)$, respectively. Then $$\align d\tilde\rho_{(\tilde\mu, \tilde\sigma),1,1}(\lambda) &= \frac{\lambda-\mu} {\lambda -\tilde\mu} \, d\rho_{1,1}(\lambda) -2 \frac{\tilde\mu-\mu}{\lambda-\tilde\mu} \, m_{-\tilde\sigma}(\tilde\mu)\, d\rho_{1,2}(\lambda) \\ &\qquad + \frac{(\tilde\mu-\mu)^2}{(\lambda-\mu)(\lambda-\tilde\mu)} \, m_{-\tilde\sigma}(\tilde\mu)^2 \, d\rho_{2,2}(\lambda), \tag 4.12 \\ d\tilde\rho_{(\tilde\mu, \tilde\sigma),1,2}(\lambda) &= d\rho_{1,2}(\lambda) - \frac{\tilde\mu -\mu}{\lambda-\mu}\, m_{-\tilde\sigma}(\tilde\mu)\, d\rho_{2,2}(\lambda), \tag 4.13 \\ d\tilde\rho_{(\tilde\mu, \tilde\sigma),2,2}(\lambda) &= \frac{\lambda-\tilde\mu} {\lambda -\mu}\, d\rho_{2,2}(\lambda), \quad \tilde\mu\neq\mu. \tag 4.14 \endalign $$ Equivalently, $$\split d\tilde\rho_{(\tilde\mu, \tilde\sigma)}(\lambda) & = (\lambda-\mu)^{-1} (\lambda-\tilde\mu)^{-1} \pmatrix \lambda -\mu & -(\tilde\mu -\mu)m_{-\tilde\sigma}(\tilde\mu) \\ 0 & \lambda - \tilde\mu \endpmatrix \\ &\qquad \times d\rho(\lambda) \pmatrix \lambda-\mu & 0 \\ -(\tilde\mu -\mu)m_{-\tilde\sigma}(\tilde\mu) & \lambda-\tilde\mu \endpmatrix, \quad \tilde\mu\neq\mu. \endsplit \tag 4.15 $$ \endproclaim The case $(\tilde\mu, \tilde\sigma)=(\mu, -\sigma)$ follows by a straightforward limiting argument (cf.~(3.12), (3.17), and (3.31)). \demo{Proof} It suffices to consider $\tilde\rho_{(\tilde\mu, \tilde\sigma),2,2}(\lambda)$, the remaining cases being analogous. Equation (4.5) and $$ \text{Im}[\tilde M_{(\tilde\mu, \tilde\sigma),2,2}(\nu+i\epsilon)] =\frac{\epsilon^2 + (\nu-\mu)(\nu-\tilde\mu)}{(\nu-\mu)^2 +\epsilon^2} \text{Im}[M_{2,2}(\nu +i\epsilon)] + \frac{(\tilde\mu -\mu)\epsilon}{(\nu-\mu)^2 +\epsilon^2} \text{Re}[M_{2,2}(\nu +i\epsilon)] $$ show that one can follow the proof of Theorem 3.5 step by step involving (3.34)--(3.36) (replacing $m_\pm (z), \rho_\pm(\lambda)$ by $M_{2,2}(z), \rho_{2,2}(\lambda)$, etc.). \qed \enddemo This finally leads to the principal spectral deformation result of this paper. \proclaim{Theorem 4.4} Assume {\rom{(H.2.3)}} and let $H$, $\tilde H_{(\tilde\mu, \tilde\sigma)}$, $H^D$, and $\tilde H^D_{(\tilde\mu, \tilde\sigma)}$ be defined by {\rom{(2.3)}}, {\rom{(2.32)}}, {\rom{(2.4)}}, and {\rom{(2.36)}}, respectively. {\rom{(i)}} Suppose $\mu,\tilde\mu\in (E_0, E_1)$. Then $\tilde H_{(\tilde\mu, \tilde\sigma)}$ and $H$ are unitarily equivalent. Moreover, $\tilde H^D_{(\tilde\mu, \tilde\sigma)}$ and $H^D$, restricted to the orthogonal complements of the one-dimensional eigenspaces corresponding to $\tilde\mu$ and $\mu$, are unitarily equivalent. {\rom{(ii)}} Assume $\mu\in\{E_0, E_1\}\cap\sigma_d (H)$, $\tilde\mu\in (E_0, E_1)$. Then $$\align \sigma_{(p)}(\tilde H_{(\tilde\mu, \tilde\sigma)}) &=\sigma_{(p)}(H)\backslash \{\mu\}, \tag 4.16 \\ \sigma_{(p)}( \tilde H^D_{(\tilde\mu, \tilde\sigma)}) &= \{\sigma_{(p)}(H^D) \backslash \{\mu\}\}\cup\{\tilde\mu\}. \tag 4.17 \endalign $$ {\rom{(iii)}} Suppose $\mu\in (E_0, E_1)$, $\tilde\mu\in \{E_0, E_1\}\cap \sigma_d (H)$. Then $$\align \sigma_{(p)}(\tilde H_{\tilde\mu}) &= \sigma_{(p)}(H)\backslash \{\tilde\mu\}, \tag 4.18 \\ \sigma_{(p)}(\tilde H^D_{\tilde\mu}) &=\sigma_{(p)}(H^D)\backslash\{\mu\}, \quad \tilde\mu\notin\sigma(\tilde H^D_{\tilde\mu}). \tag 4.19 \endalign $$ {\rom{(iv)}} Assume $\mu, \tilde\mu\in \{E_0, E_1\}\cap \sigma_d (H)$, $\mu \neq\tilde\mu$. Then $$\align \sigma_{(p)}(\tilde H_{\tilde\mu}) &= \sigma_{(p)}(H)\backslash\{E_0, E_1\}, \tag 4.20 \\ \sigma_{(p)}(\tilde H^D_{\tilde\mu}) &= \sigma_{(p)}(H^D)\backslash \{\mu\}, \quad \tilde\mu\notin\sigma (\tilde H^D_{\tilde\mu}). \tag 4.21 \endalign $$ In cases {\rom{(ii)--(iv)}}, the corresponding pairs of operators, restricted to the obvious orthogonal complements corresponding to $\mu$ and/or $\tilde\mu$, are unitarily equivalent. In particular, $$ \sigma_{\text{\rom{ess, ac, sc}}}(\tilde H_{(\tilde\mu, \tilde\sigma)}) = \sigma_{\text{\rom{ess, ac, sc}}}(\tilde H^D_{(\tilde\mu, \tilde\sigma)}) = \sigma_{\text{\rom{ess, ac, sc}}}(H^D) = \sigma_{\text{\rom{ess, ac, sc}}}(H). \tag 4.22 $$ \endproclaim \demo{Proof} This is a direct consequence of Corollary 3.3, Theorems 3.5, 3.7, and 4.4, and the orthogonal decompositions of $H^D =H^D_- \oplus H^D_+$, $\tilde H^D_{(\tilde\mu, \tilde\sigma)}= \tilde H^D_{(\tilde\mu, \tilde\sigma),-}\oplus \tilde H^D_{(\tilde\mu, \tilde\sigma),+}$. Moreover, in connection with case (iv), one observes that $\mu\in \sigma_d (H)\cap \sigma_d (H^D)$ necessarily implies that $\tilde\mu\in \{\{E_0, E_1\} \cap\sigma_d (H)\}\backslash\{\mu\}$ cannot lie in $\sigma_d (H^D)$ (i.e., two consecutive discrete eigenvalues of $H$ cannot both belong to the spectrum of $H^D$). \qed \enddemo \remark{Remark 4.5} Perhaps the most spectacular consequence of Theorem 4.4(i), from an inverse spectral point of view, is the fact that {\it{any}} finite number of deformations of Dirichlet data within spectral gaps of $V$ yields a potential $\tilde V$ in the isospectral class of $V$\!. No further constraints on $(\mu_j, \sigma_j), (\tilde\mu_j, \tilde\sigma_j)$, other than $(\mu_j, \sigma_j), (\tilde\mu_j, \tilde\sigma_j) \in (E_{j-1}, E_j) \times \{-,+\}$, $(E_{j-1}, E_j)\in\Bbb R\backslash \sigma (H)$, $j=1, \dots, N$, $N\in\Bbb N$ are involved. \endremark \smallpagebreak On an intuitive level, the Dirichlet deformation method amounts to the following two-step procedure. In the first commutation step, effected by $\psi_\sigma (\mu, x)$ in (2.19), the Dirichlet eigenvalue $\mu\in (E_0, E_1)$ associated with $H=-\frac{d^2}{dx^2} +V$ on the interval $(x_0, \sigma\infty)$ for some $x_0 =x_0 (\mu)\in\Bbb R$ is moved to $\infty$, thereby producing a singular intermediate potential deformation of $V(x)$ in the process. The second commutation step, effected by $\psi_{-\tilde\sigma} (\tilde\mu, x)$ in (2.19), then moves back this ``Dirichlet eigenvalue" from $\infty$ to $\tilde\mu \in (E_0, E_1)$ associated with the interval $(x_0, \tilde\sigma \infty)$. In the latter process, the resulting deformation $\tilde V_{(\tilde\mu, \tilde\sigma)}(x)$ becomes regular again (i.e., $W_{(\tilde\mu, \tilde\sigma)}(x)\neq 0$, $x\in\Bbb R$) and isospectral to the original base potential $V(x)$. We conclude this section with a series of remarks. A variety of additional results and possible extensions in connection with the Dirichlet deformation method will be presented in Section 5. \remark{Remark 4.6} (i) The isospectral property (i) in Theorem 4.4, in the special case of periodic potentials $V(x)$, has first been proven by Finkel, Isaacson, and Trubowitz [\fit]. Further results can be found in Buys and Finkel [\bufi] and Iwasaki [\iwa] (see also [\mckcpam]). Similar constructions in connection with Schr\"odinger operators on a compact interval can be found in P\"oschel and Trubowitz [\potr] and Ralston and Trubowitz [\ratr] (see our discussion in the introduction). (ii) By inspection, Dirichlet deformations produce the commuting diagram \vskip 0.1in $$ \matrix & (\mu_2, \sigma_2) &\\ \arrowne& &\arrowse\\ \llap{$(\mu_1$}, \sigma_1) & \arrowl & (\mu_3, \rlap{$\sigma_3)$} \endmatrix $$ \vskip 0.1in \flushpar for $(\mu_j, \sigma_j)\in [E_0, E_1] \times \{-, +\}$, $1\leq j\leq 3$ according to (H.2.3). (iii) Let $\mu\in (E_0, E_1)$. Then the (isospectral) Dirichlet deformation $(\mu,\sigma) \to (\mu, -\sigma)$ is precisely the isospectral case of the double commutation method considered in Appendix B (see Remark B.3(i)). It simply flips the Dirichlet eigenvalue $\mu$ on the half-line $(x_0, \sigma\infty)$ to the other half-line $(x_0, -\sigma\infty)$. In the special case where $V(x)$ is periodic, this procedure has first been used by McKean and van Moerbeke [\mckmoe]. (iv) In analogy to Remark 3.6, the Dirichlet deformation method as displayed in (4.12)--(4.14) changes magnitudes of discontinuities of $\rho(\lambda)$ at all eigenvalues $\lambda_n \in\sigma_p (H)$ as long as $\tilde\mu\neq\mu$. Even in the special case $(\tilde\mu, \tilde\sigma)=(\mu, -\sigma)$ discussed in item (iii) above, one obtains invariance of the magnitudes of jumps at $\lambda_n$ only for the spectral matrix element $\rho_{2,2}(\lambda)$. (v) In the non-isospectral cases (ii)--(iv), a combination of the present Dirichlet deformation method with the double commutation method in Appendix B can restore isospectrality by inserting an eigenvalue at $\mu$, $\tilde\mu$, or both. \endremark \remark{Remark 4.7} In certain cases where the base (background) potential $V$ is reflectionless (cf.~(5.6)) and $H$ is bounded from below and has no singularly continuous spectrum, the isospectral class $\text{Iso}(V)$ of $V$ (the set of all $\tilde V$'s such that $\sigma(\tilde H)=\sigma(H)$) is completely characterized by the distribution of Dirichlet (initial) data $(\mu_{j+1}(x_0), \sigma_{j+1}(x_0))\in [E_j, E_{j+1}] \times \{-,+\}$, $j\in J$ in non-trivial spectral gaps of $H$. Here $x_0\in\Bbb R$ is a fixed reference point and $J=\{0,1,\dots, N-1\}$, $N\in\Bbb N$ or $j\in J=\Bbb N_0$ ($=\Bbb N\cup\{0\}$) parametrizes these non-trivial spectral gaps $(E_j, E_{j+1})$ of $H$ (the trivial one being $(-\infty, \inf \sigma(H))$). Prime examples of this type are periodic potentials, algebro-geometric quasi-periodic finite-gap potentials, and certain limiting cases thereof (e.g., soliton potentials). In these cases, an iteration of the Dirichlet deformation method, in the sense that $(\mu_{j+1}(x_0), \sigma_{j+1} (x_0))\to (\tilde\mu_{j+1}(x_0), \tilde\sigma_{j+1}(x_0))$ within $[E_j, E_{j+1}]\times \{-, +\}$ for each $j\in J$, independently of each other (cf.~(5.4), (5.5)) yields an explicit realization of the underlying isospectral class $\text{Iso}(V)$ of potentials with base $V$\!. In the periodic case, this has first been proved by Finkel, Isaacson, and Trubowitz [\fit] (see also [\bufi], [\iwa]). More precisely, the inclusion of limiting cases $\mu_{j+1}(x_0)\in \{E _j, E_{j+1}\} \cap \sigma_{\text{\rom{ess}}}(H)$ requires a special argument (since it is excluded by (H.2.3)) but this can be provided in the special cases at hand. \endremark \remark{Remark 4.8} Another case of primary interest concerns potentials $V$ with purely discrete spectra bounded from below, that is, $$\gather \sigma(H)=\sigma_d(H) = \{E_j\}_{j\in\Bbb N_0}, \quad -\infty< E_0, \ E_j 0, \ k>0, \ q \text{ real-valued}, \\ &\quad -\infty\leq a0 ,\tag 5.13 $$ one obtains $$ f_\pm (\lambda, x) =\frac{1}{T(\lambda)}\, \overline{f_\mp (\lambda, x)} \, + \frac{R\Sp\ell \\ r\endSp (\lambda)}{T(\lambda)}\, f_\mp (\lambda, x), \quad \lambda >0 \tag 5.14 $$ and $$\aligned T(\lambda) &= \frac{2i\lambda^{1/2}} {W(f_- (\lambda), f_+ (\lambda))}\, , \\ R^\ell (\lambda) &= -\frac{W(\overline{f_- (\lambda)}, f_+ (\lambda))} {W(f_- (\lambda), f_+ (\lambda))}\, , \quad R^r (\lambda) = -\frac {W(f_- (\lambda), \overline{f_+ (\lambda)})}{W(f_- (\lambda), f_+ (\lambda))}\, , \quad \lambda >0 \endaligned \tag 5.15 $$ define the scattering matrix $S(\lambda)$ in $\Bbb C^2$ associated with the pair $(H, H_0)$, where $H_0 =-\frac{d^2}{dx^2}$, $\Cal D(H_0) =H^{2,2}(\Bbb R)$, $$ S(\lambda) =\pmatrix T(\lambda) & R^r (\lambda) \\ R^\ell (\lambda) & T(\lambda) \endpmatrix , \quad \lambda >0. \tag 5.16 $$ (5.12) and (5.14) then yield (see, e.g. [\detr], Section 2) $$\aligned f_\pm (\lambda, x) &= \cases e^{\pm i\lambda^{1/2}x} + o(1), & x\to\pm\infty \\ \frac {1}{T(\lambda)} e^{\pm i\lambda^{1/2}x} + \frac{ R\Sp \scriptscriptstyle \ell \\ \scriptscriptstyle r\endSp (\lambda)} {T(\lambda)} e^{\mp i\lambda^{1/2}x} +o(1), & x\to\mp\infty, \endcases , \\ f'_\pm (\lambda, x) &= \cases \pm i\lambda^{1/2} e^{\pm i\lambda^{1/2}x} + o(1), & x\to\pm\infty \\ \frac{\pm i\lambda^{1/2}}{T(\lambda)} e^{\pm i\lambda^{1/2}x} \mp i\lambda^{1/2} \frac{R\Sp \scriptscriptstyle \ell \\ \scriptscriptstyle r\endSp (\lambda)}{T(\lambda)} e^{\mp i\lambda^{1/2}x} +o(1), & x\to\mp\infty \endcases, \quad \lambda > 0. \endaligned \tag 5.17 $$ The following result proves that DDM preserves the class of $L^1 (\Bbb R; (1+|x|)\,dx)$ potentials and computes the scattering matrix $\tilde S_{(\tilde\mu, \tilde\sigma)}(\lambda)$, $$ \tilde S_{(\tilde\mu, \tilde\sigma)}(\lambda)= \pmatrix \tilde T_{(\tilde\mu, \tilde\sigma)}(\lambda) & \tilde R^r_{(\tilde\mu, \tilde\sigma)} (\lambda)\\ \tilde R^\ell_{(\tilde\mu, \tilde\sigma)}(\lambda) & \tilde T_{(\tilde\mu, \tilde\sigma)} (\lambda)\endpmatrix, \quad \lambda >0, \tag 5.18 $$ associated with the pair $(\tilde H_{(\tilde\mu, \tilde\sigma)}, H_0)$ in terms of $S(\lambda)$ in (5.16) associated with $(H, H_0)$. \proclaim{Lemma 5.5} In addition to {\rom{(H.2.3)}}, assume $\mu, \tilde\mu\in (E_0, E_1)\subset (-\infty,0)$. Then $\tilde V_{(\tilde\mu, \tilde\sigma)} \in L^1 (\Bbb R; (1+|x|)\,dx)$ if and only if $V\in L^1 (\Bbb R; (1+|x|)\,dx)$ and the scattering matrices $\tilde S_{(\tilde\mu, \tilde\sigma)}(\lambda)$ and $S(\lambda)$ in {\rom{(5.18)}} and {\rom{(5.16)}} associated with $(\tilde H_{(\tilde\mu, \tilde\sigma)}, H_0)$ and $(H, H_0)$, respectively, are related via $$\aligned \tilde T_{(\tilde\mu, \tilde\sigma)}(\lambda) &= T(\lambda), \\ \tilde R_{(\tilde\mu, \tilde\sigma)}\Sp\ell \\ r\endSp (\lambda) &= \frac{\lambda^{1/2} \mp i\tilde\sigma (-\tilde\mu)^{1/2}}{\lambda^{1/2}\pm i\tilde\sigma (-\tilde\mu)^{1/2}} \, \frac{\lambda^{1/2} \pm i\sigma (-\mu)^{1/2}}{\lambda^{1/2}\mp i\sigma (-\mu)^{1/2}} \, R\Sp\ell \\ r\endSp (\lambda), \quad \lambda >0. \endaligned \tag 5.19 $$ \endproclaim \demo{Proof} First, we prove that $V\in L^1 (\Bbb R; (1+|x|)\,dx)$ if and only if $\tilde V_{(\tilde\mu, \tilde\sigma)}\in L^1 (\Bbb R; (1+|x|)\, dx)$ for $\mu\neq \tilde\mu$. We adopt the strategy of Deift and Trubowitz [\detr] in their proof of Theorem 3.2 (which treats the analog of Lemma 5.5 in the single commutation context; see Appendix A and especially the paragraph preceding (A.32), (A.33)). Introduce $$ g_\sigma (\mu, x) =\psi_\sigma (\mu, x) e^{\sigma(-\mu)^{1/2}x}, \quad g_{-\tilde\sigma} (\tilde\mu, x) =\psi_{-\tilde\sigma} (\tilde\mu, x) e^{-\tilde\sigma(-\mu)^{1/2}x}, \quad \mu\neq\tilde\mu. \tag 5.20 $$ Then Lemmas 2.1 and 2.6 of [\detr] yield $$ g_\sigma (\mu, x) =C_{\sigma,\pm} (\mu) (1+o(1)), \quad x\to\pm\infty, \tag 5.21 $$ with $C_{\sigma, \pm}(\mu)>0$ and $$\gathered g'_\sigma (\mu, x) = o(1), \quad |x|\to \infty, \\ g'_\sigma (\mu, \,\cdot\, )\in L^\infty (\Bbb R) \cap L^1 (\Bbb R; (1+|x|)\,dx) \endgathered \tag 5.22 $$ and similarly for $g_{-\tilde\sigma}(\tilde\mu, x)$. Next, abbreviating $W(x)= W(\psi_\sigma (\mu), \psi_{-\tilde\sigma}(\tilde\mu))(x)$, one computes using (5.21), $$\align \tilde V_{(\tilde\mu, \tilde\sigma)} -V &=2W^{-2} [W^{\prime 2}-WW''] \\ &= \{[\tilde\sigma (-\tilde\mu)^{1/2} +\sigma (-\mu)^{1/2}] g_\sigma g_{-\tilde\sigma} + g_\sigma g'_{-\tilde\sigma}-g_{-\tilde\sigma}g'_\sigma\}^{-2} \tag 5.23 \\ &\quad \times 2(\tilde\mu-\mu) \{2\tilde\sigma (-\tilde\mu)^{1/2} g^2_\sigma g_{-\tilde\sigma}g'_{-\tilde\sigma} +2 \sigma(-\mu)^{1/2} g_\sigma g^2_{-\tilde\sigma} g'_\sigma +g^2_\sigma g^{\prime 2}_{-\tilde\sigma} -g^2_{-\tilde\sigma} g^{\prime 2}_\sigma\}, \\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \mu\neq\tilde\mu. \endalign $$ By (5.21) and (5.22), the right-hand side of (5.23) is clearly in $L^1 (\Bbb R; (1+|x|)\,dx)$ at least as long as $(\tilde\mu, \tilde\sigma)\neq(\mu, -\sigma)$. The case $(\tilde\mu, \tilde\sigma)=(\mu, -\sigma)$ leads to a rather cumbersome $0/0$ expression in (5.23). Fortunately, this is quite irrelevant since we can simply apply DDM twice, that is, use the deformation sequence $(\mu, \sigma)\to (\tilde\mu, -\sigma)\to (\mu, -\sigma)$ with $\tilde\mu\neq\mu$ (instead of $(\mu, \sigma)\to (\mu, -\sigma)$ in one step) by appealing to Remark 4.6(ii). This then proves $\tilde V_{(\tilde\mu, \tilde\sigma)} \in L^1 (\Bbb R; (1+|x|)\,dx)$ if and only if $V\in L^1 (\Bbb R; (1+|x|)\,dx)$ in all cases. Next, define $$ \tilde f_{(\tilde\mu, \tilde\sigma), \pm}(\lambda, x) = (U_{(\tilde\mu, \tilde\sigma), \pm} (\lambda) f_\pm (\lambda))(x), \quad \lambda >0, \ \tilde\mu\neq\mu $$ (cf.~(3.1)). Then (5.17) yields $$\aligned \tilde f_{(\tilde\mu, \tilde\sigma), \pm}(\lambda, x) &= \frac{\lambda^{1/2} \pm i\tilde\sigma (-\tilde\mu)^{1/2}}{\lambda^{1/2}\pm i\sigma (-\mu)^{1/2}} [e^{\pm i\lambda^{1/2}x} + o(1)], \quad x\to\pm\infty, \\ \tilde f_{(\tilde\mu, \tilde\sigma), \pm}(\lambda, x) &=\frac{\lambda^{1/2} \pm i\tilde\sigma (-\tilde\mu)^{1/2}}{\lambda^{1/2}\pm i\sigma (-\mu)^{1/2}} \biggl[\frac{1}{T(\lambda)}\, e^{\pm i\lambda^{1/2}x} + \frac{R\Sp\ell \\ r\endSp (\lambda)}{T(\lambda)} \\ &\qquad \times \frac{\lambda^{1/2} \mp i\tilde\sigma (-\tilde\mu)^{1/2}}{\lambda^{1/2} \pm i\tilde\sigma (-\tilde\mu)^{1/2}} \, \frac{\lambda^{1/2} \pm i\sigma (-\mu)^{1/2}}{\lambda^{1/2}\mp i\sigma (-\mu)^{1/2}} \, e^{\mp i\lambda^{1/2}x} +o(1)\biggr], \ x\to\mp\infty \\ &= \frac{\lambda^{1/2} \pm i\tilde\sigma (-\tilde\mu)^{1/2}}{\lambda^{1/2}\pm i\sigma (-\mu)^{1/2}} \biggl[ \frac{1}{\tilde T_{(\tilde\mu, \tilde\sigma)}(\lambda)} \, e^{\pm i\lambda^{1/2}x} + \frac{\tilde R_{(\tilde\mu, \tilde\sigma)}\Sp\ell \\ r\endSp (\lambda)} {\tilde T_{(\tilde\mu, \tilde\sigma)}(\lambda)}\, e^{\mp i\lambda^{1/2}x} + o(1)\biggr ], \\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad x\to\mp\infty, \ \lambda >0, \ \tilde\mu\neq \mu, \endaligned \tag 5.24 $$ with $\tilde T_{(\tilde\mu, \tilde\sigma)}(\lambda)$, $\tilde R_{(\tilde\mu, \tilde\sigma)}\Sp\ell \\ r\endSp (\lambda)$ as given by (5.19). Applying this two-step procedure to $\tilde S_{(\tilde\mu, \tilde\sigma)}(\lambda)$ then proves (5.19) also in the remaining (sign flip) case $(\mu, \sigma) \to (\mu, -\sigma)$. \qed \enddemo For simplicity we only considered the case $\mu, \tilde\mu\in (E_0, E_1)$ in Lemma 5.5 There is no problem in moving $\tilde\mu$ to the boundary of the interval $(E_0, E_1)$ as long as the boundary point in question is an eigenvalue of $H$ (i.e., different from $E_1 =0$). Indeed, in the case $(\mu,\sigma)\to \tilde\mu\in \{E_0, E_1\}\cap \sigma_d (H)$, $\mu\in (E_0, E_1)$, the analog of (5.19) then reads $$\aligned \tilde T_{\tilde\mu}(\lambda) &= \frac{\lambda^{1/2} -i(-\tilde\mu)^{1/2}} {\lambda^{1/2} + i(-\tilde\mu)^{1/2}}\, T(\lambda), \\ \tilde R_{\tilde\mu}\Sp \ell \\ r\endSp (\lambda) &= \frac{\lambda^{1/2} -i(-\tilde\mu)^{1/2}} {\lambda^{1/2} +i(-\tilde\mu)^{1/2}}\, \frac{\lambda^{1/2}\pm i\sigma(-\tilde\mu)^{1/2}} {\lambda^{1/2}\mp -i\sigma (-\tilde\mu)^{1/2}}\, R\Sp \ell \\ r\endSp (\lambda), \\ &\qquad \lambda>0, \ (\mu,\sigma)\in (E_0, E_1) \times \{-,+\}, \ \tilde\mu\in\{E_0, E_1\}\cap\sigma_d(H). \endaligned \tag 5.25 $$ One observes that the transmission coefficient in (5.19) stays invariant with respect to DDM (since Lemma 5.5 describes the isospectral case $\sigma_{(p)}(\tilde H_{(\tilde\mu, \tilde\sigma)})=\sigma_{(p)}(H)$ as $\mu,\tilde\mu\in (E_0, E_1)$) whereas (5.25) exhibits a change of $T(\lambda)$ (as it must since now $\tilde\mu\in \{E_0, E_1\} \cap\sigma_d(H)$ got knocked out of the spectrum of $\tilde H_{(\tilde\mu, \tilde\sigma)}$, $\sigma_{(p)} (\tilde H_{(\tilde\mu, \tilde\sigma)})=\sigma_{(p)}(H)\backslash \{\tilde\mu \}$). In this context we invite the reader to compare with the corresponding single and double commutation results in Appendices A and B (see (A.32), (A.33), and (B.35), (B.36)). \remark{Remark 5.6} It should be pointed out at the end that the material in this paper is not at all confined to Schr\"odinger or Sturm-Liouville operators. In fact, (a generalization of) Lemma 2.2 for general second-order finite-difference (Jacobi) operators appeared in [\te]. Moreover, the discrete analog of DDM was used in [\gtdiff] to construct an explicit realization of the isospectral torus for algebro-geometric quasi-periodic Jacobi operators. A detailed analysis of the discrete version of DDM will be given in [\tpre]. \endremark \vskip 0.3in \flushpar {\bf Acknowledgments.} F.G.~is indebted to C.W.~Peck and A.~Kechris for kind invitations to Caltech during the summers of 1994 and 1996 where some of this work was done. The extraordinary hospitality and support by the Department of Mathematics at Caltech are gratefully acknowledged. \vskip 0.3in \flushpar{\bf Appendix A. The Single Commutation or Crum-Darboux Method} \vskip 0.1in We briefly summarize the main spectral characteristics of the single commutation\linebreak method (abbreviated occasionally as SCM in this appendix). The principal source for the following material is the fundamental paper by Deift [\deift] (see also [\detr], [\gsvi], [\schm]). Further hints to the literature and to applications of this method in spectral theory and completely integrable systems are collected at the end of this appendix. Suppose $$ V\in L^1_{\text{\rom{loc}}}(\Bbb R)\text{ is real valued} \tag A.1 $$ and assume that the differential expression $$ \tau =-\frac{d^2}{dx^2}+V(x), \quad x\in\Bbb R\text{ is non-oscillatory at both $\pm\infty$}. \tag A.2 $$ Denote by $H$ the uniquely associated self-adjoint operator in $L^2(\Bbb R)$, maximally defined, that is, $$\aligned Hf &=\tau f, \\ f\in\Cal D(H) &= \{g\in L^2(\Bbb R)\mid g, g'\in AC_{\text{\rom{loc}}}(\Bbb R); \tau g\in L^2 (\Bbb R)\}. \endaligned \tag A.3 $$ Because of (A.2), $H$ is bounded from below, $$ H\geq \Sigma_0, \quad \Sigma_0 =\inf\sigma(H)>-\infty. \tag A.4 $$ Next, let $\psi_\pm (z,x)$, $z\in\Bbb C\backslash\sigma_{\text{\rom{ess}}}(H)$ be the unique (up to constant multiples) solutions of $$ \tau\psi (z) =z\psi(z) \tag A.5 $$ satisfying for all $z\in\Bbb C\backslash\sigma_{\text{\rom{ess}}}(H)$, $R\in\Bbb R$, $$ \psi_\pm (z, ,\cdot\,) \in L^2 ((R, \pm\infty)) \text{ and } \psi_\pm (\lambda, x) >0 \text{ for } \lambda < \Sigma_0. \tag A.6 $$ (The latter condition in (A.6) can always be achieved since $\psi_\pm (\lambda, x)\neq 0$ for $\lambda < \Sigma_0$.) One defines $$ \psi_{\nu_1} (\lambda_1,x) =\frac{1}{2} (1-\nu_1)\psi_- (\lambda_1,x) + \frac{1}{2} (1+\nu_1)\psi_+ (\lambda_1,x), \quad \nu_1 \in [-1, 1], \ \lambda_1 <\Sigma_0 \tag A.7 $$ (we identify $\psi_{\pm 1} =\psi_\pm$ for notational convenience) and $$\align \phi_{\nu_1}(\lambda_1, x) &=\psi'_{\nu_1} (\lambda_1, x) / \psi_{\nu_1} (\lambda_1, x), \tag A.8 \\ \alpha_{\nu_1}(\lambda_1) &=\frac{d}{dx} +\phi_{\nu_1} (\lambda_1, x), \quad \alpha_{\nu_1}(\lambda_1)^+ = -\frac{d}{dx} + \phi_{\nu_1} (\lambda_1, x). \tag A.9 \endalign $$ (We note that $\psi_{\nu_1} (\lambda_1, x)\neq 0$ for $\lambda_1 <\Sigma_0$). One infers that $$ \tau=\alpha_{\nu_1}(\lambda_1) \alpha_{\nu_1} (\lambda_1)^+ + \lambda_1 =-\frac{d^2}{dx^2}+V(x) \tag A.10 $$ is independent of $\nu_1\in [-1,1]$ and $\lambda_1 <\Sigma_0$ and introduces a commutation of $\alpha_{\nu_1}(\lambda_1)$ and $\alpha_{\nu_1} (\lambda_1)^+$, $$\aligned \tilde\tau_{\nu_1}(\lambda_1) &= \alpha_{\nu_1}(\lambda_1)^+ \alpha_{\nu_1}(\lambda_1)+\lambda_1 = -\frac{d^2}{dx^2} + \tilde V_{\nu_1} (\lambda_1, x), \quad x\in\Bbb R,\\ \tilde V_{\nu_1}(\lambda_1, x) &= V(x) -2\{\ln [\psi_{\nu_1} (\lambda_1, x)]\}'', \quad \lambda_1 <\Sigma_0, \ \nu_1\in [-1, 1]. \endaligned \tag A.11 $$ One verifies $$\gathered \alpha_{\nu_1}(\lambda_1)^+ \psi_{\nu_1}(\lambda_1) =0, \quad \alpha_{\nu_1}(\lambda_1) \psi_{\nu_1}(\lambda_1)^{-1} =0, \\ \psi_\pm (\lambda_1, \,\cdot\,)\in L^2 ((R, \pm\infty)), \quad \psi_\pm (\lambda_1, \,\cdot\,)\notin L^2 ((R, \mp\infty)),\quad R\in\Bbb R ,\\ \psi_{\nu_1}(\lambda_1, \,\cdot\,)\notin L^2 ((-\infty, R))\cup L^2 ((R, \infty)), \quad R\in \Bbb R, \\ \psi_{\nu_1}(\lambda_1, \,\cdot\,)^{-1}\in L^2 (\Bbb R), \quad \nu_1 \in (-1, 1). \endgathered \tag A.12 $$ Next, let $\overline{A_{\nu_1}(\lambda_1)}$, $\overline{A_{\nu_1}(\lambda_1)}^*$, and $\tilde H_{\nu_1}(\lambda_1)$ be the following closed linear operators in $L^2 (\Bbb R)$ associated with $\alpha_{\nu_1}(\lambda_1)$, $\alpha_{\nu_1} (\lambda_1)^+$, and $\tilde\tau_{\nu_1}(\lambda_1)$, respectively. Consider $$\gather \Cal D_0 = \{g\in\Cal D(H)\mid \text{ supp$(g)$ compact}\}, \tag A.13 \\ A_{\nu_1}(\lambda_1) = \left. \alpha_{\nu_1}(\lambda_1)\right|_{\Cal D_0}. \tag A.14 \endgather $$ Then, $$ \left. A_{\nu_1}(\lambda_1)^*\right|_{\Cal D_0} = \left. \alpha_{\nu_1} (\lambda_1)^+ \right|_{\Cal D_0} \tag A.15 $$ and (cf.~[\gzh]) $$ H=\overline{A_{\nu_1}(\lambda_1)} \ \overline{A_{\nu_1}(\lambda_1)}^* +\lambda_1, \quad \tilde H_{\nu_1}(\lambda_1) = \overline{A_{\nu_1}(\lambda_1)}^* \, \overline{A_{\nu_1}(\lambda_1)} + \lambda_1. \tag A.16 $$ In particular, by Lemma 5.1(i), $\tilde\tau_{\nu_1}(\lambda_1)$ (and, of course, $\tau$) is l.p.~at $\pm\infty$ since $\tilde H_{\nu_1}(\lambda_1)\geq\lambda_1$ ($H\geq\Sigma_0$). Let $$\gathered \overline{A_{\nu_1}(\lambda_1)}^* = S_{\nu_1}(\lambda_1) |\overline{A_{\nu_1} (\lambda_1)}^*| = |\overline{A_{\nu_1}(\lambda_1)}| \, S_{\nu_1} (\lambda_1), \\ |\overline{A_{\nu_1}(\lambda_1)}^*| = [\overline{A_{\nu_1}(\lambda_1)} \ \overline{A_{\nu_1}(\lambda_1)}^*]^{1/2}, \quad |\overline{A_{\nu_1}(\lambda_1)}| = [\overline{A_{\nu_1}(\lambda_1)}^* \ \overline{A_{\nu_1}(\lambda_1)}]^{1/2} \endgathered \tag A.17 $$ denote the polar decomposition of $\overline{A_{\nu_1}(\lambda_1)}^*$, where $$ S_{\nu_1}(\lambda_1): L^2 (\Bbb R) \to \text{Ker} (\,\overline{A_{\nu_1}(\lambda_1)}\,)^\perp \tag A.18 $$ is unitary. At this point we used the fact that $\text{Ker}(\overline{A_{\nu_1}(\lambda_1)}^*)= \{0\}$ by the hypothesis $\lambda_1 <\Sigma_0 =\inf \sigma(H)$. Moreover, we introduce the orthogonal projection $\tilde P_{\nu_1}(\lambda_1)$ onto $\text{Ker} (\overline{A_{\nu_1}(\lambda_1)})$, that is, $$ \tilde P_{\nu_1}(\lambda_1) =\cases 0, & \nu_1 \in \{-1, 1\} \\ \| \psi_{\nu_1}(\lambda_1)^{-1}\|^{-2}_2 (\psi_{\nu_1}(\lambda_1)^{-1}, \,\cdot\,)\psi_{\nu_1}(\lambda_1)^{-1}, & \nu_1 \in (-1, 1) \endcases \tag A.19 $$ (cf.~(A.12)). The fundamental result regarding the spectra of $H$ and $\tilde H_{\nu_1} (\lambda_1)$ then follows as a special case of the unitary equivalence of $\overline A \, \overline A^*$ and $\overline A^* \, \overline A$, restricted to the orthogonal complements of their respective null spaces (see Deift [\deift]). \proclaim{Theorem A.1 (Deift [\deift], see also [\gsvi], [\schm])} Let $H, \tilde H_{\nu_1}(\lambda_1)$, $\nu_1\in [-1,1]$, $\lambda_1 <\Sigma_0 =\inf\sigma(H)$, and $S_{\nu_1}(\lambda_1)$, $\tilde P_{\nu_1}(\lambda_1)$ be given as in {\rom{(A.3)}}, {\rom{(A.16)}}, and {\rom{(A.18)}}, {\rom{(A.19)}}. Then $$ \tilde H_{\nu_1}(\lambda_1) \restriction \text{\rom{Ran}}(1 -\tilde P_{\nu_1}(\lambda_1)) = S_{\nu_1}(\lambda_1) H S _{\nu_1}(\lambda_1)^{-1}, \tag A.20 $$ that is, $\tilde H_{\pm 1}(\lambda_1)$ and $H$, and $\tilde H_{\nu_1}(\lambda_1)$ and $H$, $\nu_1 \in (-1, 1)$, restricted to the orthogonal complement of the \rom(one-dimensional\rom) eigenspace associated with the eigenvalue $\lambda_1$ of $\tilde H_{\nu_1}(\lambda_1)$, are unitarily equivalent. In particular, $$\align \sigma_{(p)} (\tilde H_{\nu_1}(\lambda_1)) &= \cases \sigma_{(p)}(H), & \nu_1 \in \{-1, 1\} \\ \sigma_{(p)}(H)\cup \{\lambda_1\}, &\nu_1 \in (-1, 1) \endcases , \\ \text{\rom{Ker}}(\tilde H_{\nu_1}(\lambda_1)-\lambda_1) &= \cases \{0\}, & \nu_1 \in \{-1, 1\} \\ {\text{\rom{span}}}\{\psi_{\nu_1}(\lambda_1)^{-1}\}, & \nu_1 \in (-1, 1) \endcases , \tag A.21 \\ \sigma_{\text{\rom{ess, ac, sc}}} (\tilde H_{\nu_1}(\lambda_1)) &= \sigma_{\text{\rom{ess, ac, sc}}}(H). \endalign $$ \endproclaim Next, we describe a variety of additional results and possible extensions paralleling Section 5. This is intended to facilitate comparisons with DDM as well as the double commutation method in Appendix B. One verifies that $$ \tilde\psi_{\nu_1, \pm}(z,\lambda_1,x) = W(\psi_\pm (z), \psi_{\nu_1}(\lambda_1)) (x) / \psi_{\nu_1}(\lambda_1, x) \tag A.22 $$ satisfies $$\gathered \tilde\tau_{\nu_1}(\lambda_1)\tilde\psi_{\nu_1, \pm}(z,\lambda_1) = z\tilde\psi_{\nu_1, \pm}(z,\lambda_1), \\ \tilde\psi_{\nu_1, \pm}(z, \lambda_1, \,\cdot\,) \in L^2 ((R, \pm\infty)), \quad z\in\Bbb C\backslash \sigma(H), \ R\in\Bbb R. \endgathered \tag A.23 $$ (The latter fact is proven in [\gsvi], Appendix A for $z\leq \lambda_1$; one can use (2.16) to extend it to $z\in\Bbb C\backslash\sigma(H)$.) Hence, normalizing $\psi_\pm (z,x)$ temporarily (and without loss of generality) by $$ \psi_\pm (\lambda_1, x_0) =1, \quad x_0 \in\Bbb R\text{ fixed}, \tag A.24 $$ the Weyl-Titchmarsh $m$-function $\tilde m_{\nu_1, \pm}(z, \lambda_1)$ associated with $\tilde H_{\nu_1}(\lambda_1)$ and the reference point $x_0$ is given by (cf.~[\gnow]) $$\gathered \tilde m_{\nu_1, \pm}(z, \lambda_1) = \frac{z-\lambda_1} {\cot(\alpha_{\nu_1}(\lambda_1)) - m_\pm (z)} - \cot(\alpha_{\nu_1} (\lambda_1)), \\ \cot(\alpha_{\nu_1}(\lambda_1)) = \frac{1}{2} (1-\nu_1) m_- (\lambda_1) +\frac{1}{2} (1+\nu_1) m_+ (\lambda_1), \quad \nu_1 \in [-1,1], \ z\in\Bbb C\backslash\Bbb R. \endgathered \tag A.25 $$ Here use has been made of $$ \tilde m_{\nu_1, \pm}(z, \lambda_1) = \tilde\psi'_{\nu_1, \pm} (z, \lambda_1, x_0), $$ (A.22), (A.24), and the fact that $\tilde\tau_{\nu_1}(\lambda_1)$ is l.p.~at $\pm\infty$. Given (A.25) one can compute $\tilde M_{\nu_1}(z,\lambda_1)$, the Weyl-Titchmarsh $M$-matrix in $\Bbb C^2$ associated with $\tilde H_{\nu_1}(\lambda_1)$ in terms of $M(z)$, the $M$-matrix of $H$ (see (4.1)). One obtains $$\align \tilde M_{\nu_1,1,1}(z, \lambda_1)&= \frac{\cot^2 (\alpha_{\nu_1} (\lambda_1))} {z-\lambda_1} M_{1,1}(z) + 2\cot (\alpha_{\nu_1}(\lambda_1)) \biggl[ 1- \frac{\cot^2(\alpha_{\nu_1}(\lambda_1))}{z-\lambda_1}\biggr] M_{1,2} (z) \\ &\qquad + \biggl[ (z-\lambda_1) -2\cot^2 (\alpha_{\nu_1}(\lambda_1)) + \frac{\cot^4 (\alpha_{\nu_1}(\lambda_1))}{z-\lambda_1}\biggr] M_{2,2}(z), \tag A.26 \\ \tilde M_{\nu_1,1,2}(z,\lambda_1) &= -\frac{\cot(\alpha_{\nu_1} (\lambda_1))} {z-\lambda_1} M_{1,1}(z) + \biggl[ \frac{2\cot^2 (\alpha_{\nu_1} (\lambda_1))} {z-\lambda_1} -1\biggr] M_{1,2}(z), \tag A.27 \\ \tilde M_{\nu_1, 2,2}(z, \lambda_1) &= \frac{1}{z-\lambda} M_{1,1} (z) - \frac{2\cot(\alpha_{\nu_1}(\lambda_1))}{z-\lambda_1} M_{1,2}(z) + \frac{\cot^2 (\alpha_{\nu_1}(\lambda_1))}{z-\lambda_1} M_{2,2}(z), \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad z\in\Bbb C\backslash\Bbb R. \tag A.28 \endalign $$ One readily confirms that all matrix elements $\tilde M_{\nu_1,p,q} (z,\lambda_1)$, $1\leq p, q\leq 2$ have a pole at $z=\lambda_1$ if and only if $\nu_1 \in (-1, 1)$ (i.e., if and only if $\cot (\alpha_{\nu_1}(\lambda_1))\neq m_\pm (\lambda_1$) in agreement with Theorem A.1. Moreover, we might note that $m_- (\lambda_1) \neq m_+ (\lambda_1)$ since $\lambda_1 <\Sigma_0 =\inf\sigma(H)$. One could use (A.26)--(A.28) to compute the corresponding $\Bbb C^2$-valued spectral matrix $\tilde\rho_{\nu_1}(\lambda, \lambda_1)$ of $\tilde H_{\nu_1}(\lambda_1)$ in terms of $\rho(\lambda)$, the one associated with $H$. The resulting formulas (although providing an alternative proof of Theorem A.1), however, are rather complex and hence omitted. (A.26)--(A.28) become simpler if the Dirichlet boundary condition $\psi (x_0\pm 0)=0$, used to compute $\tilde m_{\nu_1, \pm}(z,\lambda_1)$, $\tilde M_{\nu_1}(z,\lambda_1)$, is replaced by $\sin (\alpha_{\nu_1} (\lambda_1)) \psi' (x_0\pm 0) +\cos (\alpha_{\nu_1}(\lambda_1)) \psi (x_0 \pm 0) =0$. We will not pursue this now but return to this approach in Appendix B. Iterations of SCM can be handled as follows. Assume $V\in L^1_{\text{\rom{loc}}} (\Bbb R)$ to be real-valued and pick $\lambda_1 <\lambda_2 <\cdots <\lambda_N <\Sigma_0=\inf\sigma(H)$, $\nu_j \in [-1, 1]$, $C_{\pm, j} >0$, $1\leq j\leq N$, $N\in\Bbb N$. Then the SCM result after $N$ iteration steps, denoted by $\tilde\tau_{\nu_1,\dots,\nu_N}(\lambda_1, \dots, \lambda_N)$, reads as follows. $$\gather \tilde\tau_{\nu_1, \dots, \nu_N}(\lambda_1,\dots, \lambda_N) = - \frac{d^2}{dx^2} +\tilde V_{\nu_1,\dots, \nu_N}(\lambda_1, \dots, \lambda_N, x), \quad x\in\Bbb R, \tag A.29 \\ \tilde V_{\nu_1, \dots, \nu_N}(\lambda_1, \dots, \lambda_N,x) = V(x) -2 \{\ln [W(\psi_{\nu_1}(\lambda_1), \dots, \psi_{\nu_N}(\lambda_N))(x)]\}'', \\ \psi_{\nu_j}(\lambda_j, x) =\frac{1}{2} (1-\nu_j)C_{-,j} \psi_- (\lambda_j, x) +\frac{1}{2} (1+\nu_j) C_{+, j} \psi_+ (\lambda_j, x),\quad 1\leq j\leq N. \endgather $$ The obvious analog of Theorem A.1 (distinguishing between $\nu_j \in (-1, 1)$ and $\nu_j \in \{-1, 1\}$) then applies to $\tilde H_{\nu_1, \dots, \nu_N} (\lambda_1, \dots, \lambda_N)$, the unique semi-bounded self-adjoint operator associated with $\tilde\tau_{\nu_1,\dots,\nu_N}(\lambda_1, \dots, \lambda_N)$ (see [\gsvi], Appendix A for more details). In analogy to Lemma 5.4, one infers from (A.25) that $\tilde H_{\nu_1}(\lambda_1)$ is reflectionless if and only if $H$ is (as observed in [\gnow]). In order to obtain the Sturm-Liouville generalization of (A.1)--(A.16) (see [\gtproc], [\schm]), one assumes $$\gathered p,p', k,k'\in AC_{\text{\rom{loc}}} ((a,b), \quad q\in L^1_{\text{\rom{loc}}} ((a,b)) \text{ real-valued}, \\ p>0, \ k>0, -\infty\leq a < b \leq \infty, \endgathered \tag A.30 $$ and makes the changes $$\gathered \alpha_{\nu_1}(\lambda_1) \to \widehat\alpha_{\nu_1}(\lambda_1) = \frac{1}{k(x)} \biggl(\sqrt{k(x)p(x)} \frac{d}{dx} + \phi_{\nu_1}(\lambda_1, x) \biggr), \\ \alpha_{\nu_1}(\lambda_1)^+ \to \widehat\alpha_{\nu_1}(\lambda_1)^+ = \frac{1}{k(x)} \biggl(-\frac{d}{dx} \sqrt{k(x)p(x)}\, +\phi_{\nu_1}(\lambda_1,x) \biggr), \\ \tau\to\widehat\tau = \widehat\alpha_{\nu_1}(\lambda_1) \widehat\alpha_{\nu_1}(\lambda_1)^+ +\lambda_1 =\frac{1}{k(x)} \biggl( - \frac{d}{dx}\, p(x) \frac{d}{dx}+ q(x)\biggr), \\ \tilde\tau_{\nu_1}(\lambda_1) \to \widetilde{\widehat\tau}_{\nu_1}(\lambda_1) =\widehat\alpha_{\nu_1}(\lambda_1)^+ \widehat\alpha_{\nu_1}(\lambda_1) +\lambda_1 =\frac{1}{k(x)} \biggl( -\frac{d}{dx}\, p(x) \frac{d}{dx} + \tilde q_{\nu_1}(\lambda_1, x)\biggr), \\ \tilde q_{\nu_1}(\lambda_1,x) = q(x) -\frac{p''(x)}{2} + \frac{p'(x)^2}{4p(x)} + \frac{3k'(x)^2 p(x)}{4k(x)^2} - \frac{k''(x)p(x)}{2k(x)} \\ \qquad \qquad \qquad + \biggl( \frac{1}{k(x)} (k(x)p(x))' -2 \frac{d}{dx}\, p(x)\biggr) \frac{d}{dx} \, \ln [\psi_{\nu_1}(\lambda_1, x)], \\ \widehat\tau \psi_{\nu_1}(\lambda_1) = \lambda_1 \psi_{\nu_1}(\lambda_1), \quad \phi_{\nu_1}(\lambda_1,x) = \bigl(\sqrt{k(x)p(x)}\, \psi_{\nu_1}(\lambda_1, x)\bigr)' \big/ \psi_{\nu_1}(\lambda_1, x). \endgathered \tag A.31 $$ It remains to sketch the scattering theory formulas analogous to (5.19), assuming $V\in L^1 (\Bbb R; (1+|x|)\,dx)$ to be real-valued. (We use the conventions established in (5.11)--(5.17).) It was proved by Deift and Trubowitz ([\detr], Theorem 3.2) that $\tilde V_{\nu_1}(\lambda_1)\in L^1 (\Bbb R; (1+|x|)\,dx)$ if and only if $V$ is, and also the scattering matrix $\tilde S_{\nu_1}(\lambda, \lambda_1)$ associated with the pair $(\tilde H_{\nu_1}(\lambda_1), H_0)$ in terms of $S(\lambda)$ in (5.16), the one corresponding to $(H, H_0)$, was determined as follows. $$\aligned \tilde T_{\nu_1}(\lambda, \lambda_1) &= \frac{\lambda^{1/2} + i(-\lambda_1)^{1/2}}{\lambda^{1/2} - i(-\lambda_1)^{1/2}}\, T(\lambda), \\ \tilde R_{\nu_1}\Sp \ell \\ r\endSp (\lambda, \lambda_1) &=- \frac{\lambda^{1/2} + i(-\lambda_1)^{1/2}}{\lambda^{1/2} - i(-\lambda_1)^{1/2}}\, R\Sp \ell \\ r\endSp (\lambda), \quad \lambda >0, \ \lambda_1 < \Sigma_0, \ \nu_1 \in (-1, 1), \endaligned \tag A.32 $$ $$\aligned \tilde T_{\nu_1}(\lambda, \lambda_1) &= T(\lambda), \\ \tilde R_{\nu_1}\Sp\ell \\ r\endSp (\lambda, \lambda_1) &=- \frac{\lambda^{1/2} \pm i\nu_1 (-\lambda_1)^{1/2}} {\lambda^{1/2} \mp i\nu_1 (-\lambda_1)^{1/2}}\, R\Sp \ell \\ r\endSp (\lambda), \quad \lambda >0, \ \lambda_1 <\Sigma_0, \ \nu_1\in \{-1, 1\}. \endaligned \tag A.33 $$ Further generalizations of (A.32), (A.33) in the context of supersymmetric quantum mechanics can be found in [\bggss] and [\gss]. The discrete analog of SCM for general second-order finite-difference (Jacobi) operators has been developed in detail in [\gtdiff]. Finally we provide a brief historical account and hints to some applications of SCM. SCM, or as it is often called, the Crum-Darboux method, (A.7)--(A.11) goes back at least to Jacobi [\jac] and Darboux [\dar]. Important later contributions are due to Crum [\crum], Schmincke [\schm], and especially, Deift [\deift] (see also [\detr]). In particular, the spectral deformation results of this method, as summarized in Theorem A.1, are due to Deift [\deift]. In recent years, this method has been applied to a description of the isospectral manifold of periodic and algebro-geometric quasi-periodic finite-gap solutions of the (m)KdV hierarchy (see, e.g, [\ehkn], [\erfl], [\gdekker], [\gcam], [\gss], [\gwei], [\mckcpam], [\mckrev], [\mckjsp], [\mckspm], [\mcknew], and the references therein) and the construction of soliton solutions (resp., reflectionless potentials) of the (m)KdV hierarchy relative to given background (base) solutions (resp., potentials) by means of B\"acklund transformations (cf., e.g., [\deift], [\detr], [\flmcl], [\gdekker], [\gsvi], [\gss], [\mckjsp], and the literature cited therein). As is obvious from (A.11), $\psi_{\nu_1}(\lambda_1, x)$ had to be chosen zero-free (and hence $\lambda_1 <\Sigma_0$) in order to guarantee $V_{\nu_1}(\lambda_1) \in L^1_{\text{\rom{loc}}}(\Bbb R)$. This considerable restriction on the range of $\lambda_1$ will be overcome in the following appendix. \vskip 0.3in \flushpar {\bf Appendix B. The Double Commutation Method} \vskip 0.1in We review the double commutation method (occasionally abbreviated as DCM) to insert eigenvalues into spectral gaps of general background (base) Schr\"odinger operators following [\gjfa] and [\gtproc]. Applications of this method and pertinent references to the literature are collected at the end of this appendix. Assuming $V$ satisfies $$ V\in L^1_{\text{\rom{loc}}}(\Bbb R), \quad V \text{ real-valued} \tag B.1 $$ and introducing the differential expression $\tau=-\frac{d^2}{dx^2} + V(x)$, $x\in \Bbb R$, we pick $\lambda_0\in\Bbb R$ and $\eta_\pm (\lambda_0, x)$ satisfying $$\gathered \tau\psi(\lambda_0) =\lambda_0 \psi(\lambda_0), \\ \eta_\pm (\lambda_0, \,\cdot\,) \in L^2 ((R, \pm\infty)), \quad R\in\Bbb R, \ \eta_\pm (\lambda_0, x)\text{ real-valued}. \endgathered \tag B.2 $$ Given $\eta_\pm (\lambda_0, x)$, we define the self-adjoint background (base) operator $H_\pm$ in $L^2(\Bbb R)$ via $$\aligned H_\pm f &=\tau f, \\ f\in\Cal D(H_\pm ) &= \{g\in L^2 (\Bbb R)\mid g,g'\in AC_{\text{\rom{loc}}}(\Bbb R); \tau g\in L^2 (\Bbb R); \\ & \qquad \lim\limits_{x\to\omega\infty} W(\eta_\pm (\lambda_0), g) (x) =0 \text{ if $\tau$ is l.c.~at $\omega \infty, \ \omega \in\{-,+\}$}\}. \endaligned \tag B.3 $$ Our choice of notation purposely stresses a possible dependence of $H_\pm$ on $\eta_\pm (\lambda_0, x)$. If $\tau$ is in the l.p.~case at $\omega \infty$, $\omega \in \{-,+\}$, the corresponding boundary condition in (B.3) is superfluous at $\omega \infty$ and hence to be deleted from (B.3). In particular, if $\tau$ is l.p.~at $\pm\infty$, then $H_- = H_+ =H$ is independent of the choice of $\eta_\pm (\lambda_0, x)$. Next, denote $$ L^2_{\text{\rom{loc}}} ([\mp\infty, \pm\infty)) = \{g\in L^2_{\text{\rom{loc}}} (\Bbb R)\mid g\in L^2 ((\mp\infty, c))\} $$ for some $c\in\Bbb R$ and pick $\gamma_1 \in (0,\infty)$, $\lambda_1\in\Bbb R$ and $\psi_\pm (z,x)$ satisfying $$\gathered \tau\psi(z) =z\psi(z), \\ \psi_\pm (z, \,\cdot\,) \in L^2 ((R,\pm\infty)), \quad z\in\Bbb C\backslash \sigma_{\text{\rom{ess}}} (H_\pm), \ R\in\Bbb R, \\ \psi_\pm (\lambda, x) \text{ real-valued for }\lambda\in\Bbb R, \\ \psi_\pm (\lambda_1, \,\cdot\,) \in L^2 ((R,\pm\infty)), \quad R\in\Bbb R, \\ \lim\limits_{x\to\omega\infty} W(\eta_\pm (\lambda_0), \psi_\pm (\lambda_1))(x) = 0 \text{ if $\tau$ is l.c.~at $\omega\infty, \ \omega\in \{-, +\}$}. \endgathered \tag B.4 $$ As in (B.3), the last condition in (B.4) is superfluous at $\omega\infty$ if $\tau$ is l.p.~at $\omega\infty$, $\omega\in\{-,+\}$. Given $\gamma_1 >0$ and $\psi_\pm (\lambda_1, x)$, we define the linear transformation $$ \widehat U_{\pm, \gamma_1}(\lambda_1) : \cases L^2_{\text{\rom{loc}}} ([\pm\infty, \mp\infty)) \to L^2_{\text{\rom{loc}}} ([\pm\infty, \mp\infty)) \\ f(x) \to \tilde f_{\pm, \gamma_1} (\lambda_1, x) = f(x) \pm \gamma_1 \tilde\psi_{\pm, \gamma_1} (\lambda_1, x) \int^x_{\pm\infty} dx' \psi_\pm (\lambda_1, x') f(x') \endcases , \tag B.5 $$ where $$ \tilde\psi_{\pm, \gamma_1} (\lambda_1, x) = (\widehat U_{\pm, \gamma_1} (\lambda_1) \psi_\pm (\lambda_1))(x) = \biggl[ 1\mp\gamma_1 \int\limits^x_{\pm\infty} dx' \, \psi_{\pm} (\lambda_1, x')^2\biggr]^{-1} \psi_{\pm} (\lambda_1, x). \tag B.6 $$ By inspection, one infers for the inverse transformation $$ \widehat U_{\pm, \gamma_1} (\lambda_1)^{-1} : \cases L^2_{\text{\rom{loc}}} ([\pm\infty, \mp\infty)) \to L^2_{\text{\rom{loc}}} ([\pm\infty, \mp\infty)) \\ h(x) \to h(x) \mp \gamma_1 \psi_\pm (\lambda_1, x) \int^x_{\pm\infty}dx' \tilde\psi_{\pm, \gamma_1}(\lambda_1, x') h(x') \endcases . \tag B.7 $$ We list a few more facts (cf.~[\gtproc]) which explain Lemma B.1 and Theorem B.2 below. $$ 1\pm \gamma_1 \int\limits^x_{\pm\infty} dx' \, \tilde\psi_{\pm, \gamma_1} (\lambda_1, x')^2 = \biggl[1\mp\gamma_1 \int\limits^x_{\pm\infty} dx' \, \psi_\pm (\lambda_1, x')^2\biggr]^{-1}, \tag B.8 $$ $$ \tilde\psi_{\pm, \gamma_1}(\lambda_1) \in L^2 (\Bbb R), \ \|\tilde\psi_{\pm, \gamma_1} (\lambda_1)\|^2_2 = \gamma^{-1}_1 \biggl\{ 1-\lim\limits_{x\to\mp\infty} \biggl[ 1\mp \gamma_1 \int\limits^x_{\pm\infty} dx' \, \psi_\pm (\lambda_1, x')^2\biggr]^{-1}\biggr\}, \tag B.9 $$ $$\multline \mp\int\limits^x_{\pm\infty} dx' \, \overline{\tilde f_{\pm, \gamma_1}(\lambda_1, x')} \, \tilde g_{\pm, \gamma_1}(\lambda_1, x') =\mp\int\limits^x_{\pm\infty} dx' \, \overline{f(x')}\, g(x') \\ \pm\gamma_1 \biggl[1\mp\gamma_1 \int\limits^x_{\pm\infty} dx' \, \psi_\pm (\lambda_1, x')^2\biggr]^{-1} \int\limits^x_{\pm\infty} dx' \, \psi_\pm (\lambda_1, x') \overline{f(x')} \int\limits^x_{\pm\infty} dx'' \, \psi_\pm (\lambda_1, x'')g(x''), \endmultline \tag B.10 $$ $$\multline \mp\int\limits^x_{\pm\infty} dx'\, \overline{f(x')}\, g(x') = \mp\int\limits^x_{\pm\infty} dx' \, \overline{\tilde f_{\pm, \gamma_1}(\lambda_1, x')}\, \tilde g_{\pm, \gamma_1}(\lambda_1, x') \\ \mp \gamma_1 \biggl[ 1\pm\gamma_1 \int\limits^x_{\pm\infty} dx' \, \tilde\psi_{\pm, \gamma_1}(\lambda_1, x')^2\biggr]^{-1} \int\limits^x_{\pm\infty} dx' \, \tilde\psi_{\pm, \gamma_1}(\lambda_1, x')\, \overline{\tilde\psi_{\pm, \gamma_1} (\lambda_1, x')} \\ \times \int\limits^x_{\pm\infty} dx''\, \tilde\psi_{\pm, \gamma_1} (\lambda_1, x'') \tilde g_{\pm, \gamma_1}(\lambda_1, x''). \endmultline \tag B.11 $$ Next, we denote the restriction of $\widehat U_{\pm, \gamma_1} (\lambda_1)$ to $L^2 (\Bbb R)$ by $$ \left. U_{\pm, \gamma_1}(\lambda_1)= \widehat U_{\pm, \gamma_1} (\lambda_1) \right|_{L^2 (\Bbb R)}, \tag B.12 $$ define the orthogonal projections $$\aligned P_\pm (\lambda_1) &=\cases 0, &\psi_\pm (\lambda_1)\notin L^2 (\Bbb R) \\ \|\psi_\pm (\lambda_1)\|^{-2}_2 (\psi_\pm (\lambda_1), \,\cdot\,) \psi_\pm (\lambda_1), & \psi_\pm (\lambda_1)\in L^2 (\Bbb R) \endcases , \\ \tilde P_{\pm, \gamma_1}(\lambda_1) &= \|\tilde\psi_{\pm, \gamma_1} (\lambda_1)\|^{-2}_2 (\tilde\psi_{\pm, \gamma_1}(\lambda_1), \,\cdot\,) \tilde\psi_{\pm, \gamma_1}(\lambda_1), \endaligned \tag B.13 $$ and introduce the double commutation differential expression $$\aligned \tilde\tau_{\pm, \gamma_1}(\lambda_1) &= -\frac{d^2}{dx^2}+ \tilde V_{\pm, \gamma_1} (\lambda_1,x), \quad x\in\Bbb R, \\ \tilde V_{\pm, \gamma_1}(\lambda_1,x) &= V(x) -2\biggl\{ \ln \biggl[ 1\mp\gamma_1 \int\limits^x_{\pm\infty} dx' \, \psi_\pm (\lambda_1, x')^2 \biggr]\biggr\}''. \endaligned \tag B.14 $$ Relations (B.5)--(B.11) then yield \proclaim{Lemma B.1 [\gtproc]} \roster \item"\rom{(i)}" $U_{\pm, \gamma_1}(\lambda_1): (1-P_\pm (\lambda_1)) L^2 (\Bbb R) \to (1-\tilde P_{\pm, \gamma_1}(\lambda_1))L^2 (\Bbb R)$ is unitary. \item"\rom{(ii)}" $\tilde\tau_{\pm, \gamma_1}(\lambda_1) (\widehat U_{\pm, \gamma_1} (\lambda_1)f) = \widehat U_{\pm, \gamma_1}(\lambda_1) (\tau f)$. \endroster \endproclaim Lemma B.1(ii) shows that $U_{\pm, \gamma_1}(\lambda_1)$ are transformation operators for the pairs \linebreak $(\tilde H_{\pm, \gamma_1}(\lambda_1), H_\pm )$ in the terminology of [\levbook], Chapter 1, [\mar], Chapter 1, that is, they map solutions of $\tau\psi(z)=z\psi (z)$, $z\in\Bbb C\backslash \{\lambda_1\}$ into those of $\tilde\tau_{\pm, \gamma_1} (\lambda_1)\tilde\psi (z) =z\tilde\psi (z)$. The self-adjoint operator $\tilde H_{\pm, \gamma_1}(\lambda_1)$ corresponding to $\tilde\tau_{\pm, \gamma_1}(\lambda_1)$ is then defined by $$\align \tilde H_{\pm, \gamma_1}(\lambda_1)f &= \tilde\tau_{\pm, \gamma_1}(\lambda_1)f, \\ f\in\Cal D(\tilde H_{\pm, \gamma_1}(\lambda_1)) &= \{g\in L^2 (\Bbb R) \mid g,g' \in AC_{\text{\rom{loc}}} (\Bbb R); \tilde\tau_{\pm, \gamma_1}(\lambda_1) g\in L^2 (\Bbb R); \tag B.15 \\ &\ \lim\limits_{x\to\omega\infty} W(\tilde\psi_{\pm, \gamma_1}(\lambda_1),g) (x)=0 \text{ if $\tilde\tau_{\pm, \gamma_1}(\lambda_1)$ is l.c.~at $\omega\infty, \ \omega\in \{-,+\}$}\}. \endalign $$ As usual, the last boundary condition at $\omega\infty$ in (B.15) is to be deleted if $\tilde\tau_{\pm, \gamma_1}(\lambda_1)$ is l.p.~at $\omega\infty$. The principal result concerning the spectra of $\tilde H_{\pm, \gamma_1}(\lambda_1)$ and $H_\pm$ then reads as follows. \proclaim{Theorem B.2 [\gtproc]} Let $H_\pm$, $\tilde H_{\pm, \gamma_1}(\lambda_1)$, $\gamma_1 >0$, $\psi_\pm (\lambda_1, x)$, $\tilde\psi_{\pm, \gamma_1}(\lambda_1,x)$, $\lambda_1 \in\Bbb R, $ and $U_{\pm, \gamma_1}(\lambda_1)$, $P_\pm (\lambda_1)$, $\tilde P_{\pm, \gamma_1}(\lambda_1)$ be given as in {\rom{(B.3)}}, {\rom{(B.15)}}, {\rom{(B.4)}}, {\rom{(B.6)}}, and {\rom{(B.12)}}, {\rom{(B.13)}}. Then {\rom{(i)}} $\lambda_1 \in \sigma_p (\tilde H_{\pm, \gamma_1}(\lambda_1))$ and $$ \text{\rom{Ker}}(\tilde H_{\pm, \gamma_1}(\lambda_1)-\lambda_1) = {\text{\rom{span}}}\{ \tilde\psi_{\pm, \gamma_1}(\lambda_1)\}. \tag B.16 $$ {\rom{(ii)}} If $\psi_\pm (\lambda_1) \notin L^2 (\Bbb R)$ and \rom(and hence $\tau$ is l.p.~at $\pm\infty$\rom), one obtains $$ \tilde H_{\pm, \gamma_1}(\lambda_1)(1-\tilde P_{\pm, \gamma_1} (\lambda_1)) = U_{\pm, \gamma_1}(\lambda_1)HU_{\pm, \gamma_1}(1-\tilde P_{\pm, \gamma_1} (\lambda_1)), \tag B.17 $$ that is, $\tilde H_{\pm, \gamma_1}(\lambda_1)$ and $H$, restricted to the orthogonal complement of the \rom(one-dimensional\rom) eigenspace associated with the eigenvalue $\lambda_1$ of $\tilde H_{\pm, \gamma_1}(\lambda_1)$, are unitarily equivalent. In particular, $$ \sigma_{(p)}(\tilde H_{\pm, \gamma_1}(\lambda_1)) = \sigma_{(p)}(H)\cup \{\lambda_1\}. \tag B.18 $$ {\rom{(iii)}} If $\psi_\pm (\lambda_1) \in L^2 (\Bbb R)$, then there exists a unitary operator $\tilde U_{\pm, \gamma_1}(\lambda_1)= \tilde U_{\pm, \gamma_1}(\lambda_1) \oplus \sqrt{1+\gamma_1 \|\psi_\pm (\lambda_1)\|^2_2}\, 1$ on $(1-P_\pm (\lambda_1)) L^2 (\Bbb R) \oplus P_\pm (\lambda_1) L^2 (\Bbb R)$ such that $$ \tilde H_{\pm, \gamma_1}(\lambda_1) = \tilde U_{\pm, \gamma_1} (\lambda_1)H_\pm \tilde U_{\pm, \gamma_1}(\lambda_1)^{-1}. \tag B.19 $$ Moreover, $$ \sigma_{\text{\rom{ess, ac, sc}}} (\tilde H_{\pm, \gamma_1} (\lambda_1)) = \sigma_{\text{\rom{ess, ac, sc}}} (H_\pm ) \tag B.20 $$ in cases {\rom{(ii)}} and {\rom{(iii)}}. \endproclaim \remark{Remark B.3} (i) Thus far we considered the case $0<\gamma_\pm <\infty$. The limit $\gamma_\pm \to\infty$ in (B.14) (implying $\tilde\psi_{\pm, \gamma_1}(\lambda_1,x) \underset {\gamma_1\to\infty} \to\longrightarrow 0$ and hence $\tilde P_{\pm, \gamma_1} (\lambda_1)=0$) formally seems to yield an isospectral deformation of $H_\pm$ when compared with Theorem B.2. One computes $$\aligned \tilde\tau_{\pm, \infty }(\lambda_1) &= -\frac{d^2}{dx^2} + \tilde V_{\pm, \infty} (\lambda_1,x), \quad x\in \Bbb R, \\ \tilde V_{\pm, \infty }(\lambda_1, x) &= V(x) -2\biggl\{\ln \biggl[\mp \int\limits^x_{\pm\infty} dx' \, \psi_\pm (\lambda_1, x')^2 \biggr]\biggr\}''. \endaligned \tag B.21 $$ A quick look at (2.19) and (2.20) then shows that (B.21) is precisely the (sign flip) Dirichlet deformation (cf.~Remark 4.6(iii)) where $$ (\lambda_1, \pm) \to (\lambda_1, \mp), \quad \tilde\tau_{\pm, \infty}(\lambda_1) = \tilde\tau_{(\lambda_1, \mp)}, \ \mu=\tilde\mu=\lambda_1, \ \tilde\sigma =-\sigma =\mp. \tag B.22 $$ As a consequence of (B.22), Cases II and III in (2.34) and (2.35) coincide and the boundary conditions in $\tilde H_{(\lambda_1, \mp)}$ (if any) are identical to those in (B.15), upon replacing $\tilde\psi_{\pm, \gamma_1} (\lambda_1, x)$ by $\gamma_1 \tilde\psi_{\pm, \gamma_1} (\lambda_1, x)$ and formally letting $\gamma_1 \to \infty$. Thus $$ \tilde H_{\omega, \infty}(\lambda_1) = \tilde H_{(\lambda_1, -\omega)}, \quad (\lambda_1, \omega) = (\mu, \omega) = (\tilde\mu, -\omega), \ \omega\in \{-, +\} \tag B.23 $$ is the right definition for the self-adjoint operator associated with $\tilde\tau_{\omega, \infty} (\lambda_1)$ in (B.21). Hence the case $\gamma_1 =\infty$ is fully covered by Sections 3--5, and (B.21) indeed gives rise to an isospectral deformation of $H_\pm$. The isospectral nature of (B.21) has been systematically exploited in the context of B\"acklund transformations for the (m)KdV equation in [\gsvi]. (ii) If $H_\pm$ has an eigenfunction $\psi_\pm (\lambda_1) \in\Cal D(H_\pm)$ associated with the eigenvalue $\lambda_1$, one can reverse DCM and remove $\lambda_1$ upon choosing $\gamma_1= -\|\psi_\pm (\lambda_1)\|^{-2}_2$. In this case, $\tilde\psi_{\pm, \gamma_1} (\lambda_1) \notin L^2 (\Bbb R)$ and hence $\tilde\tau_{\pm, \gamma_1}(\lambda_1)$ is l.p.~at $\pm\infty$ (cf.~[\gjfa], [\gtproc]). (iii) Similarly to DDM (and in contrast to SCM in Appendix A where we exploited that $\tau$ was non-oscillatory and hence l.p.~at $\pm\infty$), DCM does not necessarily produce a l.p.~differential expression $\tilde\tau_{\pm, \gamma_1}(\lambda_1)$ at $\mp\infty$ even if $\tau$ was l.p.~at $\mp\infty$. In fact, one can use (ii) above to construct an example where $\tau$ is l.p.~at $\mp\infty$ but $\tilde\tau_{\pm, \gamma_1}(\lambda_1)$ is l.c.~at $\mp\infty$ ([\gjfa], [\gtproc]). However, $\tilde\tau_{\pm, \gamma_1}(\lambda_1)$ is l.p.~at $\pm\infty$ if and only if $\tau$ is and $\tilde\tau_{\pm, \gamma_1}(\lambda_1)$ is l.c.~at $\mp\infty$ if $\tau$ is ([\gjfa], [\gtproc]). Of course, Lemma 5.1 immediately covers the present situation upon entering the obvious changes in notation. \endremark Next, we turn to a computation of Weyl-Titchmarsh functions associated with $\tilde H_{\pm, \gamma_1}(\lambda_1)$ in terms of those of $H_\pm$. Since some of the following results (such as (B.26)) are new, we provide a bit more details. First, some necessary notation. Let $x_0 \in\Bbb R$ be a fixed reference point and assume temporarily the notation used in Lemma 3.1. Given $\widehat H$, $\widehat V$, and $\widehat m_\pm (z)$, the corresponding $m$-functions associated with the half-line $(x_0, \pm\infty)$, we define the usual fractional linear transformation of $\widehat m_\pm (z)$, $$\aligned \widehat m^\alpha_\pm (z) &= \frac{-1 +\cot (\alpha) \widehat m_\pm (z)} {\cot(\alpha) +\widehat m_\pm (z)}\, , \quad \alpha\in (0,\pi), \\ \widehat m^0_\pm (z) &= \widehat m_\pm (z), \quad z\in\Bbb C\backslash\Bbb R \endaligned \tag B.24 $$ and $$\aligned \widehat M^\alpha (z) &= \pmatrix \format \l&\quad \r\\ \cos(\alpha) & -\sin(\alpha) \\ \sin(\alpha) & \cos(\alpha) \endpmatrix \widehat M(z) \pmatrix \format \l&\quad \r\\ \cos(\alpha) & -\sin(\alpha) \\ \sin(\alpha) & \cos(\alpha) \endpmatrix^{-1} \\ &= [\widehat m^\alpha_- (z) -\widehat m^\alpha_+ (z)]^{-1} \pmatrix \widehat m^\alpha_- (z) \widehat m^\alpha_+ (z) & \tfrac{1}{2} [\widehat m^\alpha_- (z) +\widehat m^\alpha_+ (z) ] \\ \tfrac{1}{2} [\widehat m^\alpha_- (z) +\widehat m^\alpha_+ (z)] & 1 \endpmatrix , \\ \widehat M^0 (z) &= \widehat M(z), \quad z\in\Bbb R, \endaligned \tag B.25 $$ with $\widehat M(z)$ defined in terms of $\widehat m_\pm (z)$ as in (4.1). (B.24) and (B.25) are associated with the boundary condition $\sin(\alpha)\psi' (x_0\pm 0)+\cos(\alpha) \psi (x_0 \pm 0)=0$, $\alpha\in (0,\pi)$ as opposed to the Dirichlet boundary condition $\alpha =0$, $\psi (x_0 \pm 0)=0$ in connection with $\widehat m_\pm (z)$ and $\widehat M(z)$. \proclaim{Lemma B.4} Denote by $\tilde m^\beta_{\omega, \gamma_1, \pm}(z, \lambda_1)$ and $m^\alpha_\pm (z)$ the corresponding $m$-functions for \linebreak $\tilde H_{\omega, \gamma_1} (\lambda_1)$, $\omega\in \{-,+\}$ and $H$ \rom($=H_\omega$\rom) associated with the half-line $(x_0, \pm\infty)$. \rom{(i)} Suppose $m^0_\omega (\lambda_1)\ne\infty$ (i.e., $\psi_\omega (\lambda_1, x_0)\ne0$), $\omega \in \{-,+\}$. Then $$\align &\tilde m^{\beta_\omega (\lambda_1)}_{\omega, \gamma_1, \pm}(z, \lambda_1) = \cot (\beta_\omega (\lambda_1)) \\ &\quad + \frac{\sin^2 (\alpha_\omega (\lambda_1))}{\sin^2 (\beta_\omega (\lambda_1))} \biggl[ m^{\alpha_\omega (\lambda_1)}_\pm (z) - \cot (\alpha_\omega (\lambda_1)) + \frac{\omega \tilde\gamma_1 \big( \psi_\omega (\lambda_1, x_0)^2 + \psi'_\omega(\lambda_1, x_0)^2\big) } {z-\lambda_1} \biggr], \\ \qquad\tilde\gamma_1 &=\gamma_1 \biggl[ 1-\omega\gamma_1 \int\limits^{x_0}_{\omega\infty} dx \, \psi_\omega (\lambda_1, x)^2\biggr]^{-1}, \tag B.26 \\ &\cot(\alpha_\omega (\lambda_1)) =-m^0_\omega (\lambda_1), \ \cot(\beta_\omega (\lambda_1)) = \cot(\alpha_\omega (\lambda_1)) -\omega\tilde\gamma_1 \psi_\omega (\lambda_1, x_0)^2. \endalign $$ \rom{(ii)} Suppose $m^0_\omega (\lambda_1)=\infty$ (i.e., $\psi_\omega (\lambda_1, x_0)=0$), $\omega \in \{-,+\}$. Then $$ \tilde m^0_{\omega, \gamma_1, \pm}(z, \lambda_1) = m^0_\pm (z) + \frac{\omega \tilde\gamma_1 \psi'_\omega(\lambda_1, x_0)^2 }{z-\lambda_1} , \tag B.27 $$ with $\tilde\gamma_1$ as in (B.26). \endproclaim \demo{Proof} We recall that $$ \tilde\psi_{\omega, \gamma_1,\pm}(z,\lambda_1,x) = \psi_\pm (z,x) - \omega \gamma_1 \frac{ \tilde\psi_{\omega, \gamma_1}(\lambda_1, x)}{z-\lambda_1} W( \psi_\omega (\lambda_1), \psi_\pm (z)) (x) \tag B.28 $$ satisfies $$\gathered \tilde\tau_{\omega, \gamma_1}(\lambda_1) \tilde\psi_{\omega, \gamma_1,\pm} (z, \lambda_1) = z\tilde\psi_{\omega, \gamma_1,\pm}(z,\lambda_1) , \\ \tilde\psi_{\omega, \gamma_1,\pm}(z,\lambda_1,\,\cdot\,)\in L^2 ((R,\pm\infty)), \quad z\in\Bbb C\backslash \sigma (\tilde H_{\omega, \gamma_1} (\lambda_1)), \ R\in\Bbb R \endgathered \tag B.29 $$ and also note that $$\gathered \tilde\psi_{\omega, \gamma_1,\omega}(\lambda_1, \lambda_1, x) = \tilde\psi_{\omega, \gamma_1}(\lambda_1, x) = \biggl[ 1-\omega\gamma_1 \int\limits^x_{\omega\infty} dx' \, \psi_\omega (\lambda_1, x')^2 \biggr]^{-1} \psi_\omega (\lambda_1, x), \\ \left. \frac{d}{dx} W(\psi_{\omega_1} (z_1), \psi_{\omega_2}(z_2))(x) \right|_{x=x_0} = z_1 - z_2 , \\ m^0_\omega (z) =\psi'_\omega (z, x_0) / \psi_\omega (z, x_0) , \\ W(\psi_{\omega_1}(z_1), \psi_{\omega_2}(z_2))(x_0) = m^0_{\omega_2} (z_2) - m^0_{\omega_1}(z_1). \endgathered \tag B.30 $$ (B.29) and (B.30) then yield $$\align & \tilde m^0_{\omega, \gamma_1, \pm}(z, \lambda_1) = \tilde\psi'_{\omega, \gamma_1, \pm} (z, \lambda_1, x_0) / \tilde\psi_{\omega, \gamma_1, \pm} (z, \lambda_1, x_0) \\ &\quad = \{1-\omega \tilde\gamma_1 \psi_\omega (\lambda_1, x_0)^2 (m^0_\pm (z) - m^0_\omega (\lambda_1)) (z-\lambda_1)^{-1}\}^{-1} \tag B.31 \\ &\quad\times \{m^0_\pm (z) + \omega\tilde\gamma_1 \psi_\omega (\lambda_1, x_0)^2 -\omega\tilde\gamma_1 \psi_\omega (\lambda_1, x_0)^2\\ &\quad\times [(m^0_\omega (\lambda_1) + \omega\tilde\gamma_1 \psi_\omega (\lambda_1, x_0)^2) (m^0_\pm (z) - m^0_\omega (\lambda_1))] (z-\lambda_1)^{-1}\} \quad \text{if } m^0_\omega (\lambda_1)\ne\infty \endalign $$ and (B.27) by a limiting procedure if $m^0_\omega (\lambda_1)= \infty$. Applying (B.24) to $m^0_\pm (z)$ and $\tilde m^0_{\omega, \gamma_1, \pm} (z,\lambda_1)$ with the choices $\cot(\alpha_\omega (\lambda_1)) =- m^0_\omega (\lambda_1)$ and $\cot(\beta_\omega (\lambda_1)) =\cot(\alpha_\omega (\lambda_1)) - \omega\tilde\gamma_1 \psi_\omega (\lambda_1,x_0)^2$ then shows by a straightforward (but somewhat painful) computation that (B.31) is equivalent to (B.26). \qed \enddemo The singularity structure of (B.26) and (B.27) near $z=\lambda_1$ then leads to a corresponding pole behavior of $\tilde M^{\beta_\omega (\lambda_1)}_{\omega, \gamma_1} (z, \lambda_1)$, $\tilde M^0_{\omega, \gamma_1}(z, \lambda_1)$ (the $\Bbb C^2$-valued $M$-matrices of $\tilde H_{\omega, \gamma_1}(\lambda_1)$) when compared to $M^{\alpha_\omega (\lambda_1)}(z)$, $M^0 (z)$ (the $M$-matrices of $H$). The actual expressions for the $M$-matrices, the half-line spectral functions, and the $\Bbb C^2$-valued spectral matrix of $\tilde H_{\omega, \gamma_1}(\lambda_1)$ for $\alpha_\omega \neq 0$, $\beta_\omega \neq 0$ in terms of those of $H$ are similar to the special case $\alpha_\omega = \beta_\omega =0$ and $m_{\omega_0}(\lambda_1) =\infty$ described in detail in [\gjfa]. While they provide an alternative proof of Theorem B.2, we resist the temptation of providing detailed formulas at this point. (B.26) appears to be a new result. Iterations of DCM can now be performed as follows. Assume $V\in L^1_{\text{\rom{loc}}} (\Bbb R)$ to be real-valued and pick $\omega\in \{-,+\}$, $\gamma_j >0$, $\lambda_j \in\Bbb R$, $1\leq j\leq N$, $N\in\Bbb N$. Then the DCM result after $N$ iteration steps, denoted by $\tilde\tau_{\omega, \gamma_1, \dots, \gamma_N} (\lambda_1, \dots, \lambda_N)$, reads as follows. $$\gathered \tilde\tau_{\omega, \gamma_1, \dots, \gamma_N}(\lambda_1, \dots, \lambda_N) =-\frac{d^2}{dx^2} + \tilde V_{\omega, \gamma_1, \dots, \gamma_N}(\lambda_1, \dots, \lambda_N, x), \quad x\in\Bbb R, \\ \tilde V_{\omega, \gamma_1, \dots, \gamma_N}(\lambda_1, \dots, \lambda_N, x) = V(x) -2\{\ln [\det (1+C_{\omega, N}(x))]\}'', \\ C_{\omega, N}(x) = \biggl(-\omega\gamma^{1/2}_k \gamma^{1/2}_\ell \int\limits^x_{\omega\infty} dx' \, \psi_\omega (\lambda_{k}, x')\psi_\omega (\lambda_{\ell}, x')\biggr)_{1\leq k, \ell\leq N}. \endgathered \tag B.32 $$ The analog of Theorem B.2 then applies to $\tilde H_{\omega, \gamma_1, \dots, \gamma_N}(\lambda_1, \dots, \lambda_N)$, the self-adjoint operator associated with $\tilde\tau_{\omega, \gamma_1, \dots, \gamma_N}(\lambda_1, \dots, \lambda_N)$ (defined similarly to (B.15)) as discussed in detail in [\gtproc] (see also [\gss], [\gsvi]). As in Lemma 5.4 one infers from (B.31) that $\tilde H_{\omega, \gamma_1}(\lambda_1)$ is reflectionless if and only if $H$ is (as observed in [\gnow]). The Sturm-Liouville generalization of (B.1)--(B.14) then leads to the following results. One assumes $$\gather p^{-1}, q, k\in L^1_{\text{\rom{loc}}} ((a,b)), \quad kp\in AC_{\text{\rom{loc}}} ((a,b)), \quad q \text{ real-valued}, \\ p>0, \ k>0, \quad -\infty\leq a < b \leq \infty \endgather $$ and makes the substitutions (see [\gtproc]) $$\gathered \tau\to \widehat\tau =\frac{1}{k(x)} \biggl( -\frac{d}{dx}\, p(x) \frac{d}{dx} + q(x) \biggr), \\ \tilde\tau_{\omega, \gamma_1}(\lambda_1) \to \widetilde{\widehat\tau}_{\omega, \gamma_1}(\lambda_1) = \frac{1}{k(x)} \biggl( -\frac{d}{dx}\, p(x) \frac{d}{dx} + \tilde q_{\omega, \gamma_1}(\lambda_1, x)\biggr), \\ \tilde q_{\omega, \gamma_1}(\lambda_1, x) =q(x) + \biggl(\frac{1}{k(x)} \, (k(x)p(x))' -2\frac{d}{dx}\, p(x)\biggr) \\ \qquad \qquad \times \frac{d}{dx}\, \ln \biggl[ 1-\omega\gamma_1 \int\limits^x_{\omega\infty} k(x')dx'\, \psi_\omega (\lambda_1, x')^2 \biggr], \quad \omega\in \{-, +\}. \endgathered \tag B.33 $$ It remains to sketch scattering theory for real-valued potentials $V\in L^1 (\Bbb R; (1+|x|)\,dx)$ similar to DDM and SCM in Section 5 and Appendix A. (We again use the conventions established in (5.11)--(5.17).) Suppose $\lambda_1 <0$, $\gamma_1 >0$. Then we claim that DCM leaves $L^1 (\Bbb R; (1+|x|)\,dx)$ potentials invariant as in DDM and SCM. More precisely, we assert that $\tilde V_{\omega, \gamma_1} (\lambda_1)\in L^1 (\Bbb R; (1+|x|)\,dx)$ if and only if $V\in L^1 (\Bbb R; (1+|x|)\,dx)$. Indeed, since $$\multline \tilde V_{\omega, \gamma_1}(\lambda_1,x) -V(x) = \biggl[ 1-\omega\gamma_1 \int\limits^x_{\omega\infty} dx' \, \psi_\omega (\lambda_1, x')^2\biggr] ^{-2} 2\gamma^2_1 \psi_\omega (\lambda_1, x)^4 \\ + \bigg[ 1-\omega\gamma_1 \int\limits^x_{\omega\infty} dx' \, \psi_\omega (\lambda_1, x')^2 \biggr]^{-1} 4\omega\gamma_1 \psi_\omega (\lambda_1, x) \psi'_\omega (\lambda_1, x), \quad \omega\in \{-, +\}, \endmultline \tag B.34 $$ the right-hand side of (B.34) is exponentially decreasing near $\omega\infty$ and hence in \linebreak $L^1 ((R, \omega\infty); (1+|x|)\,dx)$ for all $R\in\Bbb R$. In order to treat (B.34) near $-\omega\infty$, one expands $$\split \biggl[ \gamma^{-1}_1 -\omega \int\limits^x_{\omega\infty} dx' \psi_\omega (\lambda_1, x')^2 \biggr]^{-1} &\underset{x\to -\omega\infty} \to = \biggr(-\omega \int\limits^x_{\omega\infty} dx' \psi_\omega (\lambda_1, x')^2 \biggr)^{-1} \\ &\qquad \quad \times \biggl[ 1+O \biggl(\biggl(-\omega \int\limits^x_{\omega\infty} dx' \psi_\omega (\lambda_1, x')^2 \biggr)^{-1}\biggr)\biggr] \endsplit $$ in (B.34) and notices that $O((\dots)^{-1})$ is exponentially decreasing as $x\to -\omega\infty$. The proof of our assertion is then finished by observing that the leading order term in (B.34) is precisely the isospectral double commutation deformation corresponding to $\gamma_1 =\infty$ (cf.~Remark B.4(i)), which in turn corresponds to the (sign flip) Dirichlet deformation $(\lambda_1, +\omega) \to (\lambda_1, -\omega)$. The latter has been dealt with in Lemma 5.5. The fact that DCM leaves $L^1 (\Bbb R; (1+|x|)\,dx)$ potentials invariant was proved by Levitan [\levbook], Section 6.6 using a different strategy (which yields exponential decay of $[\tilde V_{\omega, \gamma_1}(\lambda_1, x)-V(x)]$ also as $x\to-\omega\infty$). Finally, we compute the scattering matrix in this context. Following the arguments in (5.11)--(5.20), one readily verifies the following expressions for the scattering matrix $\tilde S_{\omega, \gamma_1}(\lambda, \lambda_1)$ of the pair $(\tilde H_{\omega, \gamma_1}(\lambda_1), H_0)$ in terms of $S(\lambda)$ corresponding to $(H, H_0)$. $$\align \tilde T_{\omega, \gamma_1}(\lambda, \lambda_1) &= \frac{\lambda^{1/2} + i(-\lambda_1)^{1/2}}{\lambda^{1/2} - i(-\lambda_1)^{1/2}} \, T(\lambda), \quad \omega\in \{-,+\}, \\ \tilde R^\ell_{\omega, \gamma_1}(\lambda, \lambda_1) &= \cases R^\ell (\lambda), & \omega =- \\ \bigl(\tfrac{\lambda^{1/2} + i(-\lambda_1)^{1/2}}{\lambda^{1/2}- i(-\lambda_1)^{1/2}} \bigr)^2 R^\ell (\lambda), & \omega =+ \endcases , \tag B.35 \\ \tilde R^r_{\omega, \gamma_1}(\lambda, \lambda_1) &= \cases \bigl(\tfrac{\lambda^{1/2} + i(-\lambda_1)^{1/2}}{\lambda^{1/2}- i(-\lambda_1)^{1/2}} \bigr)^2 R^r (\lambda), & \omega =- \\ R^r (\lambda), & \omega=+ \endcases , \ \lambda >0, \ \lambda_1\in (-\infty, 0)\backslash \sigma_d (H), \ \gamma_1 >0, \\ \tilde T_{\omega, \gamma_1}(\lambda, \lambda_1) &= T(\lambda), \quad \tilde R_{\omega, \gamma_1}\Sp \ell \\ r\endSp (\lambda, \lambda_1) = R\Sp \ell \\ r\endSp (\lambda), \quad \omega\in \{-, +\}, \ \lambda >0, \, \lambda_1\in\sigma_d (H), \ \gamma_1 > 0. \tag B.36 \endalign $$ The case $\gamma_1 =\infty$ is a special case of (5.19) with $\mu=\tilde\mu= \lambda_1$, $\sigma =-\tilde\sigma =\omega$. The discrete analog of DCM for general second-order finite-difference (Jacobi) operators can be found in [\gtdiff]. As in Appendix A, we conclude with a brief account of the history of DCM and references to further applications of it. The seminal work by Gel'fand and Levitan [\gellev] in 1955 on a solution of the inverse spectral problem seems to mark the first appearance of DCM where it has been used in connection with Wigner-von Neumann examples on the half-line $(0, \infty)$. (For a more recent treatment of the half-line case $(0, \infty)$, see [\deift].) Shortly afterward, the construction of reflectionless potentials in the particular case of $\tau=-\frac{d^2}{dx^2}$ (i.e., $V\equiv 0$), using double commutation formulas as a result of applying the inverse scattering approach, was derived by Kay and Moses [\kaymo]. Their result regained prominence when Gardner, Greene, Kruskal, and Miura [\ggkm] used this formalism to solve the initial value problem for the KdV equation and derived the KdV $N$-soliton solutions. The case of background (base) potentials $V\in L^1 (\Bbb R; (1+|x|)\,dx)$ is considered at length in [\detr] and [\levbook], Section 6.6. The case of periodic finite-gap background potentials is treated in [\fir], [\flmcl], [\gsvi], and [\kumi]. Close connections between the double commutation technique and the inverse spectral method based on Marchenko's approach can be inferred, for instance, from [\fir], [\flmcl], [\ggkm], [\gss], [\kaymo], and [\kumi]. General backgrounds were first treated in [\gsvi] (see also [\eastkalf], Chapter 4). In particular, the construction of KdV and mKdV soliton solutions relative to general (m)KdV background solutions on the basis of (single and double) commutation techniques has been systematically studied in [\gsvi]. 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