SCATTERING THEORY FOR JACOBI OPERATORS WITH GENERAL STEPLIKE QUASI-PERIODIC BACKGROUND

IRYNA EGOROVA, JOHANNA MICHOR, AND GERALD TESCHL

To Vladimir Aleksandrovich Marchenko and Leonid Andreevich Pastur, our teachers and inspiring colleagues.

Abstract. We develop direct and inverse scattering theory for Jacobi operators with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. We give a complete characterization of the scattering data, which allow unique solvability of the inverse scattering problem in the class of perturbations with finite first moment.

1. Introduction

In this paper we consider direct and inverse scattering theory for Jacobi operators with steplike quasi-periodic finite-gap background, using the Marchenko [15] approach.

Scattering theory for Jacobi operators is a classical topic with a long tradition. Originally developed on an informal level by Case in [5], the first rigorous results for the case of a constant background were given by Guseinov [12] with further extensions by Teschl [19], [20]. The case of periodic backgrounds was completely solved in [24] (who in fact handle almost periodic operators with a homogenous Cantor type spectrum) respectively [8] using different approaches. Moreover, the case of a steplike situation, where the coefficients are asymptotically close to two different quasi-periodic finite-gap operators, was solved in [11] (see also [1], [7]) under the restriction that the two background operators are isospectral. It is the purpose of the present paper to remove this restriction.

We should also mention that scattering theory for Jacobi operators is directly applicable to the investigation of the Toda lattice with initial data in the above mentioned classes. See for example [3], [6], [23] for steplike constant backgrounds, and [9], [10], [13], [14], and [16] for periodic backgrounds. For further possible applications and additional references we refer to the discussion in [11].

Finally, let us give a brief overview of the remaining sections. After recalling some necessary facts on algebro-geometric quasi-periodic finite-gap operators in Section 2, we construct the transformation operators and investigate the properties of the scattering data in Section 3. In Section 4 we derive the Gel’fand-Levitan-Marchenko equation and show that it uniquely determines the operator. In addition, we formulate necessary conditions for the scattering data to uniquely determine our Jacobi operator. Our final Section 5 shows that our necessary conditions for the scattering data are also sufficient.

2000 Mathematics Subject Classification. Primary 47B36, 81U40; Secondary 34L25, 39A11.

Key words and phrases. Inverse scattering, Jacobi operators, quasi-periodic, steplike.

Work supported by the Austrian Science Fund (FWF) under Grants No. Y330 and J2655.

Zh. Mat. Fiz. Anal. Geom. 4-1, 33–62 (2008).

2. Step-like finite-band backgrounds

First we need to recall some facts on quasi-periodic finite-band Jacobi operators which contain all periodic operators as a special case. We refer to [20, Chapter 9] and [8] for details.

Let H_{q}^{±} be two quasi-periodic finite-band Jacobi operators,^{1}

(2.1) H_{q}^{±}f(n) = a_{q}^{±}(n)f(n + 1) + a_{q}^{±}(n − 1)f(n − 1) + b_{q}^{±}(n)f(n), f ∈ ℓ^{2}(ℤ),

associated with the Riemann surface of the square root

(2.2)  P_{±}(z) = − ∏_{j=0}^{2g_{±}+1} \sqrt{z − E_{j}^{±}},  E_{0}^{±} < E_{1}^{±} < ⋯ < E_{2g_{±}+1}^{±},

where g_{±} ∈ ℕ and \sqrt{.} is the standard root with branch cut along (−∞, 0). In fact,

H_{q}^{±} are uniquely determined by fixing a Dirichlet divisor ∑_{j=1}^{g^{±}}(μ_{j}^{±}, σ_{j}^{±}), where μ_{j}^{±} ∈ [E_{2j−1}^{±}, E_{2j}^{±}] and σ_{j}^{±} ∈ {−1, 1}. The spectra of H_{q}^{±} consist of g_{±} + 1 bands

(2.3)  σ_{±} := σ(H_{q}^{±}) = ⋃_{j=0}^{g_{±}}[E_{2j}^{±}, E_{2j+1}^{±}].

We will identify the set ℂ \ σ(H_{q}^{±}) with the upper sheet of the Riemann surface. The upper and lower sides of the cuts over the spectrum are denoted by σ^{u} and σ^{l} and the symmetric points on these cuts by λ^{u} and λ^{l}, that is,

f(λ^{u}) = lim_{ϵ↓0} f(λ + iϵ), f(λ^{l}) = lim_{ϵ↓0} f(λ − iϵ), λ ∈ σ_{±}.

We will develop the scattering theory for the operator

(2.4)  Hf(n) = a(n − 1)f(n − 1) + b(n)f(n) + a(n)f(n + 1), n ∈ ℤ,

whose coefficients are asymptotically close to the coefficients of H_{q}^{±} on the corresponding half-axes:

(2.5)  ∑_{n=0}^{±∞} |n|(|a(n) − a_{q}^{±}(n)| + |b(n) − b_{q}^{±}(n)|) < ∞.

The special case H_{q}^{−} = H_{q}^{+} has been exhaustively studied in [8] (see also [24]) and the case where H_{q}^{−} and H_{q}^{+} are in the same isospectral class σ_{−} = σ_{+} was treated in [11]. Several results are straightforward generalizations, in such situations we will simply refer to [8], [11] and only point out possible differences.

Let ψ_{q}^{±}(z, n) be the Floquet solutions of the spectral equations

(2.6)  H_{q}^{±}ψ(n) = zψ(n),  z ∈ ℂ,

1Everywhere in this paper the sub or super index ”+” (resp. ”−”) refers to the background on the right (resp. left) half-axis.

that decay for z ∈ ℂ \ σ_{±} as n → ±∞. They are uniquely defined by the condition ψ_{q}^{±}(z, 0) = 1, ψ_{q}^{±}(z, ·) ∈ ℓ^{2}(ℤ_{±}). The solution ψ_{q}^{+}(z, n) (resp. ψ_{q}^{−}(z, n)) coincides with the upper (resp. lower) branch of the Baker–Akhiezer functions of H_{q}^{+} (resp. H_{q}^{−}), see [20]. The second solutions ψ̆_{q}^{±}(z, n) are given by the other branch of the Baker–Akhiezer functions and satisfy ψ̆_{q}^{±}(z, ·) ∈ ℓ^{2}(ℤ_{∓}) as z ∈ ℂ \ σ_{±}. Their Wronskian is equal to

(2.7)  W_{q}^{±}(ψ̆_{q}^{±}(z), ψ_{q}^{±}(z)) = ±\frac{1}{ρ_{±}(z)},

where

(2.8)  ρ_{±}(z) = \frac{∏_{j=1}^{g_{±}}(z − μ_{j}^{±})}{P_{±}(z)}

satisfy by our choice of the branch for the square root

(2.9)  Im(ρ_{±}(λ^{u})) > 0, Im(ρ_{±}(λ^{l})) < 0, λ ∈ σ_{±}.

In (2.7) the following notation is used

(2.10)  W_{q,n}^{±}(f, g) := a_{q}^{±}(n) (f(n)g(n + 1) − f(n + 1)g(n)) .

Note that ψ_{q}^{±}(z, n), ψ̆_{q}^{±}(z, n) have continuous limits as z → λ^{u,l} ∈ σ_{±}^{u,l} \ ∂σ_{±}, where

∂σ_{±} = {E_{0}^{±}, ..., E_{2g_{±}+1}^{±}},

and they satisfy the symmetry property

(2.11)  ψ_{q}^{±}(λ^{l}, n) = \overline{ψ_{q}^{±}(λ^{u}, n)} = ψ̆_{q}^{±}(λ^{u}, n), λ ∈ σ_{±}.

The points (μ_{j}^{±}, σ_{j}^{±}), 1 ≤ j ≤ g_{±}, form the divisors of poles of the Baker– Akhiezer functions. Correspondingly, the sets of Dirichlet eigenvalues {μ_{1}^{±}, ..., μ_{g_{±}}^{±}} can be divided in three disjoint subsets

M^{±} = {μ_{j}^{±} | μ_{j}^{±} ∈ ℝ\σ_{±} is a pole of ψ_{q}^{±}(z, 1)},
M̆^{±} = {μ_{j}^{±} | μ_{j}^{±} ∈ ℝ\σ_{±} is a pole of ψ̆_{q}^{±}(z, 1)},
M̂^{±} = {μ_{j}^{±} | μ_{j}^{±} ∈ ∂σ_{±}}.

(2.12)

In order to remove the singularities of ψ_{q}^{±}(z, n), ψ̆_{q}^{±}(z, n) we introduce

\begin{aligned}δ_{±}(z) & := ∏_{μ_{j}^{±}∈M_{±}}(z − μ_{j}^{±}), \\ δ̂_{±}(z) & := ∏_{μ_{j}^{±}∈M_{±}}(z − μ_{j}^{±}) ∏_{μ_{j}^{±}∈M̂_{±}} \sqrt{z − μ_{j}^{±}}, \\ δ̆_{±}(z) & := ∏_{μ_{j}^{±}∈M̆_{±}}(z − μ_{j}^{±}) ∏_{μ_{j}^{±}∈M̂_{±}} \sqrt{z − μ_{j}^{±}}, \end{aligned} (2.13)

where ∏ = 1 if there are no multipliers, and set

(2.14)  ψ̃_{q}^{±}(z, n) = δ_{±}(z)ψ_{q}^{±}(z, n), ψ̂_{q}^{±}(z, n) = δ̂_{±}(z)ψ_{q}^{±}(z, n).

Lemma 2.1. The Floquet solutions ψ_{q}^{±}, ψ̆_{q}^{±} have the following properties:

(i) The functions ψ_{q}^{±}(z, n) (resp. ψ̆_{q}^{±}(z, n)) are holomorphic as functions of z in the domain ℂ\(σ_{±} ∪M_{±}) (resp. ℂ\(σ_{±} ∪M̆_{±})), take real values on the set ℝ \ σ_{±}, and have simple poles at the points of the set M_{±} (resp. M̆_{±}). They are continuous up to the boundary σ_{±}^{u} ∪ σ_{±}^{l} except at the points in M̂_{±} and satisfy the symmetry property (2.11). For E ∈ M̂_{±}, they satisfy

ψ_{q}^{±}(z, n) = O (\frac{1}{\sqrt{z − E}} ) , ψ̆_{q}^{±}(z, n) = O (\frac{1}{\sqrt{z − E}} ) , z → E ∈ M̂_{±}.

Moreover, the estimate

(2.15)  ψ̂_{q}^{±}(z, n) − ψ̂_{q}^{±}(E, n) = O(\sqrt{z − E}),  E ∈ ∂σ_{±},

is valid. (ii) The following asymptotic expansions hold as z → ±∞

(2.16)  ψ_{q}^{±}(z, n) = z^{∓n}( \begin{matrix}n−1 \\ ∏ & ∗ \\ j=0 \end{matrix} a_{q}^{±}(j))^{±1}(1 ± \frac{1}{z} \begin{matrix}n−1 \\ ∑ & ∗ \\ j=0 \end{matrix} b_{q}^{±}(j + {1 \atop 0}) + O(\frac{1}{z^{2}} )),

where

\begin{matrix}n−1 \\ ∏ & ∗ \\ j=n_{0} \end{matrix} f(j) = \{ \begin{matrix}∏_{j=n_{0}}^{n−1} f(j), & n > n_{0}, \\ 1, & n = n_{0}, \\ ∏_{j=n}^{n_{0}−1} f(j)^{−1}, & n < n_{0}, \end{matrix} \begin{matrix}n−1 \\ ∑ & ∗ \\ j=n_{0} \end{matrix} f(j) = \{ \begin{matrix}∑_{j=n_{0}}^{n−1} f(j), & n > n_{0}, \\ 0, & n = n_{0}, \\ − ∑_{j=n}^{n_{0}−1} f(j), & n < n_{0}. \end{matrix}

(iii) The functions ψ_{q}^{±}(λ, n) form a complete orthogonal system on the spectrum with respect to the weight

(2.17)  dω_{±}(λ) = \frac{1}{2πi}ρ_{±}(λ)dλ,

namely

(2.18)  ∮_{σ_{±}} \overline{ψ_{q}^{±}(λ, m)}ψ_{q}^{±}(λ, n)dω_{±}(λ) = δ(n, m),

where

(2.19)  ∮_{σ_{±}} f(λ)dλ := ∫_{σ_{±}^{u}} f(λ^{u})dλ − ∫_{σ_{±}^{l}} f(λ^{l})dλ.

Here δ(n, m) = 1 if n = m and δ(n, m) = 0 else is the Kronecker delta.

3. Scattering data

Now let H be a steplike operator with coefficients a(n), b(n) satisfying (2.5). The two solutions ψ_{±}(z, n) of the spectral equation

(3.1)  Hψ = zψ, z ∈ ℂ,

which are asymptotically close to the Floquet solutions ψ_{q}^{±}(z, n) of the background equations (2.6) as n → ±∞, are called Jost solutions. They can be represented as (see [8])

(3.2)  ψ_{±}(z, n) = ∑_{m=n}^{±∞} K_{±}(n, m)ψ_{q}^{±}(z, m),

where the functions K_{±}(n, .) are real valued and satisfy the estimate (3.3)

|K_{±}(n, m)| ≤ C_{±}(n) ∑_{j=[ \frac{m+n}{2} ]}^{±∞} (|a(j) − a_{q}^{±}(j)| + |b(j) − b_{q}^{±}(j)|), ±m > ±n > 0.

The functions C_{±}(n) > 0 decrease monotonically as n → ±∞. Moreover, we have

\begin{aligned}a(n) & = a_{q}^{+}(n)\frac{K_{+}(n + 1, n + 1)}{K_{+}(n, n)}, \\ a(n) & = a_{q}^{−}(n)\frac{K_{−}(n, n)}{K_{−}(n + 1, n + 1)}, \\ b(n) & = b_{q}^{+}(n) + a_{q}^{+}(n)\frac{K_{+}(n, n + 1)}{K_{+}(n, n)} − a_{q}^{+}(n − 1)\frac{K_{+}(n − 1, n)}{K_{+}(n − 1, n − 1)}, \\ b(n) & = b_{q}^{−}(n) + a_{q}^{−}(n − 1)\frac{K_{−}(n, n − 1)}{K_{−}(n, n)} − a_{q}^{−}(n)\frac{K_{−}(n + 1, n)}{K_{−}(n + 1, n + 1)}, \end{aligned} (3.4)

which implies (cf. [8]) the following asymptotic behavior of the Jost solutions as z → ±∞ using (3.2), (2.16), (3.5)

ψ_{±}(z, n) = z^{∓n}K_{±}(n, n)( \begin{matrix}n−1 \\ ∏ & ∗ \\ j=0 \end{matrix} a_{q}^{±}(j))^{±1}(1+(B_{±}(n)± \begin{matrix}n \\ ∑ & ∗ \\ j=1 \end{matrix} b_{q}^{±}(j−{0 \atop 1}))\frac{1}{z}+O(\frac{1}{z^{2}} )),

where

(3.6)  B_{±}(n) = ∑_{m=n±1}^{±∞}(b_{q}^{±}(m) − b(m)).

For λ ∈ σ_{±}^{u} ∪ σ_{±}^{l} a second pair of solutions of (3.1) is given by

(3.7)  ψ̆_{±}(λ, n) = ∑_{m=n}^{±∞} K_{±}(n, m)ψ̆_{q}^{±}(λ, m), λ ∈ σ_{±}^{u} ∪ σ_{±}^{l},

which cannot be continued to the complex plane. Note that ψ̆_{±}(λ, n) = \overline{ψ_{±}(λ, n)}, λ ∈ σ_{±}, and from (2.5), (3.2) we conclude

(3.8)  W (\overline{ψ_{±}(λ)}, ψ_{±}(λ)) = W_{q}^{±}(ψ̆_{q}^{±}(λ), ψ_{q}^{±}(λ)) = ±ρ_{±}(λ)^{−1}.

The Jost solutions ψ_{±} are holomorphic in the domains ℂ\(σ_{±} ∪ M_{±}) and inherit almost all properties of their background counterparts listed in Lemma 2.1. As before, we set

(3.9)  ψ̃_{±}(z, n) = δ_{±}(z)ψ_{±}(z, n), ψ̂_{±}(z, n) = δ̂_{±}(z)ψ_{±}(z, n).

The following Lemma is proven in [8].

Lemma 3.1. The Jost solutions have the following properties.

(i) For all n, the functions ψ_{±}(z, n) are holomorphic in the domain ℂ \ (σ_{±} ∪ M_{±}) with respect to z and continuous up to the boundary (σ_{±}^{u} ∪ σ_{±}^{l}) \ ∂σ_{±}, where

(3.10)  ψ_{±}(λ^{u}, n) = \overline{ψ_{±}(λ^{l}, n)}, λ ∈ (σ_{±}^{u} ∪ σ_{±}^{l}) \ ∂σ_{±}.

The functions ψ_{±}(z, n) are real valued for z ∈ ℝ\σ_{±} and have simple poles at μ_{j} ∈ M_{±}. Moreover, ψ̂_{±} are continuous up to the boundary σ_{±}^{u} ∪ σ_{±}^{l}.

(ii) At the band edges we have for λ ∈ σ_{±}^{u,l}

(3.11)  \begin{matrix}ψ_{±}(λ, n) − \overline{ψ_{±}(λ, n)} = o(1), & E ∈ ∂σ_{±} \ M̂_{±}, \\ ψ_{±}(λ, n) + \overline{ψ_{±}(λ, n)} = o(\frac{1}{\sqrt{λ−E}} ), & E ∈ M̂_{±}. \end{matrix}

Next, we introduce the sets

(3.12)  σ^{(2)} := σ_{+} ∩ σ_{−}, σ_{±}^{(1)} = clos (σ_{±} \ σ^{(2)}), σ := σ_{+} ∪ σ_{−},

where σ is the (absolutely) continuous spectrum of H and σ_{+}^{(1)} ∪ σ_{−}^{(1)} resp. σ^{(2)} are the parts which are of multiplicity one resp. two. We will denote the interior of the spectrum by int(σ), that is, int(σ) := σ \ ∂σ.

In addition to the continuous part, H has a finite number of eigenvalues situated in the gaps, σ_{d} = {λ_{1}, ..., λ_{p}} ⊂ ℝ \ σ (see, e.g., [18]). For every eigenvalue we introduce the corresponding norming constants

(3.13)  γ_{±,k}^{−1} = ∑_{n∈ℤ} |ψ̃_{±}(λ_{k}, n)|^{2}, 1 ≤ k ≤ p.

Moreover, ψ̃_{±}(λ_{k}, n) = c_{k}^{±}ψ̃_{∓}(λ_{k}, n) with c_{k}^{+}c_{k}^{−} = 1.

Let

(3.14)  W (z) := W (ψ_{−}(z), ψ_{+}(z))

be the Wronskian of two Jost solutions. This function is meromorphic in the domain ℂ \ σ with possible poles at the points M_{+} ∪ M_{−} ∪ (M̂_{+} ∩ M̂_{−}) and with possible square root singularities at the points M̂_{+} ∪ M̂_{−} \ (M̂_{+} ∩ M̂_{−}). Set

(3.15)  W̃ (z) = W (ψ̃_{−}(z), ψ̃_{+}(z)), Ŵ (z) = W (ψ̂_{−}(z), ψ̂_{+}(z)),

then Ŵ (λ) is holomorphic in the domain ℂ \ ℝ and continuous up to the boundary. But unlike to W (z) and W̃ (z), the function Ŵ (λ) may not take real values on the set ℝ \ σ and complex conjugated values on the different sides of the spectrum. That is why it is more convenient to characterize the spectral properties of the operator H by means of the function W̃, which can have singularities at the points of the sets M̂_{+} ∪ M̂_{−}. We will study the precise character of these singularities in Lemma 3.2 below.

Note that outside the spectrum the function W̃ (z) vanishes precisely at the eigenvalues. However, it might also vanish inside the spectrum at points in ∂σ_{−} ∪ ∂σ_{+}. We will call such points virtual levels of the operator H,

(3.16)  σ_{v} := {E ∈ σ : Ŵ (E) = 0},

and we will show that σ_{v} ⊆ ∂σ ∪ (∂σ_{+}^{(1)} ∩ ∂σ_{−}^{(1)}) in Lemma 3.2. All other points E of the set ∂σ_{+} ∪ ∂σ_{−} correspond to the generic case Ŵ (E) ≠ 0.

Our next aim is to derive the properties of the scattering matrix. Introduce the scattering relations

(3.17)  T_{∓}(λ)ψ_{±}(λ, n) = \overline{ψ_{∓}(λ, n)} + R_{∓}(λ)ψ_{∓}(λ, n), λ ∈ σ_{∓}^{u,l},

where the transmission and reflection coefficients are defined as usual,

(3.18) T_{±}(λ) := \frac{W (\overline{ψ_{±}(λ)}, ψ_{±}(λ))}{W (ψ_{∓}(λ), ψ_{±}(λ))}, R_{±}(λ) := −\frac{W (ψ_{∓}(λ), \overline{ψ_{±}(λ)})}{W (ψ_{∓}(λ), ψ_{±}(λ))}, λ ∈ σ_{±}^{u,l}.

The equalities in (3.18) imply the identity

\frac{1}{T_{+}(λ)ρ_{+}(λ)} = \frac{1}{T_{−}(λ)ρ_{−}(λ)} = W (λ), λ ∈ σ^{(2)},

where W (λ) is the Wronskian of two Jost solutions (3.14). This Wronskian plays an important role in the characterization of the properties of the scattering matrix. Namely, the following result is valid.

Lemma 3.2. The entries of the scattering matrix have the following properties:

I.

(a) T_{±}(λ^{u}) = \overline{T_{±}(λ^{l})}, λ ∈ σ_{±},
R_{±}(λ^{u}) = \overline{R_{±}(λ^{l})},  λ ∈ σ_{±},
(b)  \frac{T_{±}(λ)}{\overline{T_{±}(λ)}} = R_{±}(λ),  λ ∈ σ_{±}^{(1)},
(c) 1 − |R_{±}(λ)|^{2} = \frac{ρ_{±}(λ)}{ρ_{∓}(λ)} |T_{±}(λ)|^{2}, λ ∈ σ^{(2)},
(d) \overline{R_{±}(λ)}T_{±}(λ) + R_{∓}(λ)\overline{T_{±}(λ)} = 0, λ ∈ σ^{(2)}.

II. The functions T_{±}(λ) can be extended analytically to the domain ℂ\(σ∪M_{±} ∪ M̆_{±}) and satisfy

(3.19)  \frac{1}{T_{+}(z)ρ_{+}(z)} = \frac{1}{T_{−}(z)ρ_{−}(z)} = W (z).

The function W (z) has the following properties:

(a) The function W̃ (z) = δ_{+}(z)δ_{−}(z)W (z) is holomorphic on ℂ \ σ with simple zeros at the eigenvalues λ_{k}, where

(3.20)  (\frac{dW̃}{dz} (λ_{k}))^{2} = \frac{1}{γ_{+,k}γ_{−,k}} .

Moreover,

(3.21)  \overline{W̃ (λ^{u})} = W̃ (λ^{l}), λ ∈ σ,  W̃ (z) ∈ ℝ, z ∈ ℝ \ σ.

(b) The function Ŵ (z) = δ̂_{+}(z)δ̂_{−}(z)W (z) is continuous on the set ℂ\σ up to the boundary σ^{u}∪σ^{l}. It can have zeros on the set ∂σ∪(∂σ_{+}^{(1)}∩∂σ_{−}^{(1)}) and does not vanish at the other points of the spectrum σ. If Ŵ (E) = 0 as E ∈ ∂σ ∪ (∂σ_{+}^{(1)} ∩ ∂σ_{−}^{(1)}), then

(3.22)  \frac{1}{Ŵ (λ)} = O (\frac{1}{\sqrt{λ − E}} ) , for λ ∈ σ close to E.

Moreover,

(3.23)  \frac{1}{Ŵ (z)} = O ((z − E)^{−1/2−ε}) , for z close to E.

(c) In addition,

(3.24)  T_{+}(∞) = T_{−}(∞) > 0.

III. (a) The reflection coefficients R_{±}(λ) are continuous functions on int(σ_{±}^{u,l}).

(b) If E ∈ ∂σ_{+} ∩ ∂σ_{−} and Ŵ (E) ≠ 0, then the functions R_{±}(λ) are also continuous at E. Moreover,

(3.25)  R_{±}(E) = \{ \begin{matrix}−1 for & E ∉ M̂_{±}, \\ 1 for & E ∈ M̂_{±}. \end{matrix}

Proof. I. The symmetry property (a) follows from formulas (3.18) and (3.10). For (b), use (3.18) and observe that ψ_{∓}(λ) are real valued for λ ∈ int(σ_{±}^{(1)}). Let λ ∈ int(σ^{(2)}). By (3.17),

|T_{±}|^{2}W (ψ_{∓}, \overline{ψ_{∓}}) = (|R_{±}|^{2} − 1)W (ψ_{±}, \overline{ψ_{±}}),

and property (c) follows from (3.8). The consistency condition (d) can be derived directly from definition (3.18).

II. The identity (3.19) follows from (3.18). (a) The Wronskian inherits the properties of ψ_{±}(z), so it remains to show (3.20). If Ŵ (z_{0}) = 0 for z_{0} ∈ ℂ \ σ, then

(3.26)  ψ̃_{±}(z_{0}, n) = c^{±}ψ̃_{∓}(z_{0}, n)

for some constants c^{±} (depending on z_{0}), which satisfy c^{−}c^{+} = 1. In particular, each zero of W̃ (or Ŵ) outside the continuous spectrum is a point of the discrete spectrum of H and vice versa.

Let γ_{±,j} be the norming constants defined in (3.13) for some point of the discrete spectrum λ_{j}. By virtue of [20], Lemma 2.4,

\frac{d}{dz} W (ψ̃_{−}(z), ψ̃_{+}(z))|_{λ_{j}} = W_{n}(ψ̃_{−}(λ_{j}), \frac{d}{dz} ψ̃_{+}(λ_{j})) + W_{n}(\frac{d}{dz} ψ̃_{−}(λ_{j}), ψ̃_{+}(λ_{j}))

= − ∑_{k∈ℤ} ψ̃_{−}(λ_{j}, k)ψ̃_{+}(λ_{j}, k) = −\frac{1}{c_{j}^{±}γ_{±,j}} . (3.27)

Since c_{j}^{−}c_{j}^{+} = 1, we obtain (3.20).

(b) Continuity of Ŵ up to the boundary follows from the corresponding property of ψ̂_{±}(z, n). We begin with the investigation of the possible zeros of this function on the spectrum.

First let λ_{0} ∈ int(σ^{(2)}) := σ^{(2)} \ ∂σ^{(2)}, that is, δ̂_{−} ≠ 0 and δ̂_{+} ≠ 0. Suppose W (λ_{0}) = 0, then ψ_{+}(λ_{0}, n) = c ψ_{−}(λ_{0}, n) and \overline{ψ_{+}(λ_{0}, n)} = c̄\overline{ψ_{−}(λ_{0}, n)}, i.e. W (ψ_{+}, \overline{ψ_{+}}) = |c|^{2}W (ψ_{−}, \overline{ψ_{−}}). But this implies opposite signs for ρ_{+}, ρ_{−} by (3.8), sign ρ_{+}(λ_{0}) = − sign ρ_{−}(λ_{0}), which contradicts (2.9).

Let λ_{0} ∈ int(σ_{±}^{(1)}) and W̃ (λ_{0}) = 0. The point λ_{0} can coincide with a pole μ ∈ M_{∓}. But ψ_{±}(λ_{0}, n) and \overline{ψ_{±}(λ_{0}, n)} are linearly independent and bounded, and ψ̃_{∓}(λ_{0}, n) ∈ ℝ. If W (λ_{0}) = 0, then ψ̃_{∓} = c_{1}^{±} ψ_{±} = c_{2}^{±} \overline{ψ_{±}} which implies W (ψ_{±}, \overline{ψ_{±}})(λ_{0}) = 0, a contradiction.

In the general mutual location of the background spectra the case λ_{0} = E ∈ (∂σ^{(2)} ∩ int(σ_{±})) ⊂ int(σ) can occur. If Ŵ (E) = 0, then W (ψ_{±}, ψ̂_{∓})(E) = 0, where ψ̂_{∓} are defined by (3.9). The values of ψ̂_{∓}(E, ·) are either purely real or purely imaginary, therefore W (\overline{ψ_{±}}, ψ̂_{∓})(E) = 0, that is, \overline{ψ_{±}(E, n)} and ψ_{±}(E, n) are linearly dependent, which is impossible at inner points of the set σ_{±}.

Thus, the virtual level σ_{v} of H defined in (3.16) can only be located on the set ∂σ_{−} ∩ ∂σ_{+}, that is,

(3.28)  σ_{v} ⊆ ∂σ ∪ (∂σ_{−}^{(1)} ∩ ∂σ_{+}^{(1)}).

To prove (3.22), take E ∈ σ_{v} and assume for example E ∈ σ_{+}. By (3.17) and (3.19),

\frac{δ̂_{+}(λ)ψ̂_{−}(λ, n)}{δ̂_{−}(λ)ρ_{+}(λ)W (λ)} = δ̂_{+}(λ)\overline{ψ_{+}(λ, n)} + R_{+}(λ)ψ̂_{+}(λ, n).

Choose n_{0} such that ψ̂_{−}(E, n_{0}) ≠ 0. By continuity we also have ψ̂_{−}(λ, n_{0}) ≠ 0 in a small vicinity of E. Then

\frac{δ̂_{+}(λ)\overline{ψ_{+}(λ, n_{0})} + R_{+}(λ)ψ̂_{+}(λ, n_{0})}{ψ̂_{−}(λ, n_{0})} = O(1), λ → E.

Accordingly,

\frac{1}{Ŵ (λ)} = O(\frac{∏_{j=1}^{g_{+}}(λ − μ_{j}^{+})}{δ̂_{+}^{2}(λ)\sqrt{λ − E}}) = O (\frac{1}{\sqrt{λ − E}} ) , λ ∈ σ_{+},

which proves (3.22). To see (3.23) note that

g(z, n) = \frac{ψ_{+}(z, n)ψ_{−}(z, n)}{W (z)}

is a Herglotz function. Moreover, we can assume that μ_{j} ≠ E and choose n such that ψ_{±}(E, n_{0}) ≠ 0. Hence it remains to show the corresponding estimate for g(z) = g(z, n_{0}). Since the continuous spectrum of H is purely absolutely continuous, we deduce from Stieltjes inversion formula that

g(z) = \frac{1}{π} ∫_{E−δ}^{E+δ} \frac{Im(g(λ))}{λ − z}dλ + g̃(z), δ > 0,

where g̃(z) is holomorphic near E. By (3.22) we infer (λ − E)^{1/2+ε}Im(g(λ)) is Hölder continuous and the result follows from [17, Eq. (29.8)].

(c) Equation (3.24) follows from (3.5).

III. (a) follows from the corresponding properties of ψ_{±}(z) and from II, (b). To show III, (b) we use that by (3.18) the reflection coefficients have the representation

(3.29)  R_{±}(λ) = −\frac{W (\overline{ψ_{±}(λ)}, ψ_{∓}(λ))}{W (ψ_{±}(λ), ψ_{∓}(λ))} = ∓\frac{W (\overline{ψ_{±}(λ)}, ψ_{∓}(λ))}{W (λ)}

and are continuous on both sides of the set int(σ_{±}) \ (M_{∓} ∪ M̂_{∓}). Moreover,

|R_{±}(λ)| = |\frac{W (\overline{ψ̂_{±}(λ)}, ψ̂_{∓}(λ))}{Ŵ (λ)}|,

where the denominator does not vanish on the set σ_{±} \ σ_{v}. Hence R_{±}(λ) are continuous on this set since both numerator and denominator are.

Next, let E ∈ ∂σ_{±} \ σ_{v} (in particular Ŵ (E) ≠ 0). Then, if E ∉ M̂_{±}, we use (3.29) in the form

(3.30)  R_{±}(λ) = −1 ∓ \frac{δ̂_{±}(λ)W (ψ_{±}(λ) − \overline{ψ_{±}(λ)}, ψ̂_{∓}(λ))}{Ŵ (λ)},

which shows R_{±}(λ) → −1 since ψ_{±}(λ) − \overline{ψ_{±}(λ)} → 0 by Lemma 3.1, (2). This settles the first case in (3.25). Similarly, if E ∈ M̂_{±}, we use (3.29) in the form

(3.31)  R_{±}(λ) = 1 ± \frac{δ̂_{±}(λ)W (ψ_{±}(λ) + \overline{ψ_{±}(λ)}, ψ̂_{∓}(λ))}{Ŵ (λ)},

which shows R_{±}(λ) → 1 since δ̂_{±}(λ) = O(\sqrt{λ − E}) and ψ_{±}(λ) + \overline{ψ_{±}(λ)} = o(\frac{1}{\sqrt{λ−E}} ) by Lemma 3.1, (2). This settles the second case in (3.25) as well. □

4. The Gel’fand-Levitan-Marchenko equation

The aim of this section is to derive the inverse scattering problem equation (the Gel’fand-Levitan-Marchenko equation) and to discuss some additional properties of the scattering data which are consequences of this equation.

Theorem 4.1. The inverse scattering problem (the GLM) equation has the form

(4.1)  K_{±}(n, m) + ∑_{l=n}^{±∞} K_{±}(n, l)F_{±}(l, m) = \frac{δ(n, m)}{K_{±}(n, n)},  ±m ≥ ±n,

where

F_{±}(m, n) = ∮_{σ_{±}} R_{±}(λ)ψ_{q}^{±}(λ, m)ψ_{q}^{±}(λ, n)dω_{±}

+ ∫_{σ_{∓}^{(1),u}} |T_{∓}(λ)|^{2}ψ_{q}^{±}(λ, m)ψ_{q}^{±}(λ, n)dω_{∓} + ∑_{k=1}^{p} γ_{±,k}ψ̃_{q}^{±}(λ_{k}, n)ψ̃_{q}^{±}(λ_{k}, m). (4.2)

Proof. Consider a closed contour Γ_{ϵ} consisting of a large circular arc and some contours inside this arc, which envelope the spectrum σ at a small distance ε from the spectrum. Let ±m ≥ ±n. The residue theorem, (2.17), (3.5), (3.20), and equality ψ̃_{∓}(λ_{k}, n) = c_{j}^{∓}ψ̃_{±}(λ_{k}, n) imply

\frac{1}{2πi} ∮_{Γ_{ϵ}} \frac{ψ_{∓}(λ, n)ψ_{q}^{±}(λ, m)}{W (λ)}dλ = \frac{δ(n, m)}{K_{±}(n, n)} + ∑_{k=1}^{p} Res_{λ_{k}}(\frac{ψ̃_{∓}(λ, n)ψ̃_{q}^{±}(λ, m)}{W̃ (λ)})

= \frac{δ(n, m)}{K_{±}(n, n)} − ∑_{k=1}^{p} γ_{±,k}ψ̃_{±}(λ_{k}, n)ψ̃_{q}^{±}(λ_{k}, m), (4.3)

since the integrand is meromorphic on ℂ\σ with simple poles at the eigenvalues λ_{k} and at ∞ if m = n. It is continuous till the boundary except at the points E ∈ ∂σ_{+} ∪ ∂σ_{−} where

(4.4)  \frac{ψ_{∓}(λ, n)ψ_{q}^{±}(λ, m)}{W (λ)} = O (\frac{1}{\sqrt{λ − E}} ) , E ∈ ∂σ_{+} ∪ ∂σ_{−},

by (3.22). On the other hand, as ϵ → 0,

\begin{matrix}\frac{1}{2πi} ∮_{σ} \frac{ψ_{∓}(λ, n)ψ_{q}^{±}(λ, m)}{W (λ)}dλ = \\ = \frac{1}{2πi} ∮_{σ_{±}} \frac{(\overline{ψ_{±}(λ, n)} + R_{±}(λ)ψ_{±}(λ, n))ψ_{q}^{±}(λ, m)}{T_{±}(λ)W (λ)}dλ \\ + \frac{1}{2πi} ∮_{σ_{∓}^{(1)}} \frac{ψ_{∓}(λ, n)ψ_{q}^{±}(λ, m)}{W (λ)}dλ \\ = ∮_{σ_{±}} \overline{ψ_{±}(λ, n)}ψ_{q}^{±}(λ, m)dω^{±} + ∮_{σ_{±}} R_{±}(λ)ψ_{±}(λ, n)ψ_{q}^{±}(λ, m)dω^{±} \\ + \frac{1}{2πi} ∫_{σ_{∓}^{(1),u}} ψ_{q}^{±}(λ, m)(\frac{ψ_{∓}(λ, n)}{W (λ)} − \frac{\overline{ψ_{∓}(λ, n)}}{\overline{W (λ)}})dλ. \end{matrix} (4.5)

It remains to treat the last integrand. By (3.17) and Lemma 3.2, I,

\overline{ψ_{∓}(λ, n)} = T_{∓}(λ)ψ_{±}(λ, n) − R_{∓}(λ)ψ_{∓}(λ, n) = T_{∓}(λ)ψ_{±}(λ, n) − \frac{T_{∓}(λ)}{\overline{T_{∓}(λ)}}ψ_{∓}(λ, n),

and therefore

\frac{ψ_{∓}(n)}{W} − \frac{\overline{ψ_{∓}(n)}}{\overline{W}} = \frac{\overline{W} \overline{T_{∓}} + W T_{∓}}{|W |^{2}\overline{T_{∓}}} ψ_{∓}(n) − \frac{T_{∓}}{\overline{W}} ψ_{±}(n) = −\frac{T_{∓}}{\overline{W}} ψ_{±}(n),

since \overline{W} \overline{T_{∓}} + W T_{∓} = 2Re(W T_{∓}) = 0 on σ_{∓}. In summary, (4.3) and (4.5) yield

\frac{δ(n, m)}{K_{±}(n, n)} = K_{±}(n, m) + ∮_{σ_{±}} R_{±}(λ)ψ_{±}(λ, n)ψ_{q}^{±}(λ, m)dω^{±}

+ ∫_{σ_{∓}^{(1),u}} |T_{∓}(λ)|^{2}ψ_{±}(λ, n)ψ_{q}^{±}(λ, m)dω^{∓} + ∑_{j=1}^{p} γ_{±,j}ψ̃_{±}(λ_{j}, n)ψ̃_{q}^{±}(λ_{j}, m),

and applying (3.2) finishes the proof. □

As it is shown in [8], the estimate (3.3) for K_{±}(n, m) implies the following estimates for F_{±}(n, m).

Lemma 4.2. The kernel of the GLM equation satisfies the following properties. IV. There exist functions C_{±}(n) > 0 and q_{±}(n) ≥ 0, n ∈ ℤ_{±}, such that C_{±}(n) decay as n → ±∞, |n|q(n) ∈ ℓ^{1}(ℤ_{±}), and

\begin{matrix}|F_{±}(n, m)| ≤ C_{±}(n) ∑_{j=n+m}^{±∞} q(j), \\ ∑_{n=n_{0}}^{±∞} |n||F_{±}(n, n) − F_{±}(n ± 1, n ± 1)| < ∞, \\ ∑_{n=n_{0}}^{±∞} |n||a_{q}^{±}(n)F_{±}(n, n + 1) − a_{q}^{±}(n − 1)F_{±}(n − 1, n)| < ∞. \end{matrix} (4.6)

In summary, we have obtained the following necessary conditions for the scattering data:

Theorem 4.3. The scattering data

\begin{aligned}S & = \{R_{+}(λ), T_{+}(λ), λ ∈ σ_{+}^{u,l}; R_{−}(λ), T_{−}(λ), λ ∈ σ_{−}^{u,l}; \\ & λ_{1}, …, λ_{p} ∈ ℝ \ (σ_{+} ∪ σ_{−}), γ_{±,1}, …, γ_{±,p} ∈ ℝ_{+}\} \end{aligned} (4.7)

satisfy the properties I-III listed in Lemma 3.2. The functions F_{±}(n, m), defined in (4.2), satisfy property IV in Lemma 4.2.

In fact, the conditions on the scattering data given in Theorem 4.3 are both necessary and sufficient for the solution of the scattering problem in the class (2.5). The sufficiency of these conditions as well as the algorithm for the solution of the inverse problem will be discussed in the next section.

5. The inverse scattering problem

Let H_{q}^{±} be two arbitrary quasi-periodic Jacobi operators associated with sequences a_{q}^{±}(n), b_{q}^{±}(n) as introduced in Section 2. Let S be given scattering data with corresponding kernels F_{±}(n, m) satisfying the necessary conditions of Theorem 4.3.

First we show that the GLM equations (4.1) can be solved for K_{±}(n, m) if F_{±}(n, m) are given.

Lemma 5.1. Under condition IV, the GLM equations (4.1) have unique real-valued solutions K_{±}(n, ·) ∈ ℓ^{1}(n, ±∞) satisfying the estimates

(5.1)  |K_{±}(n, m)| ≤ C_{±}(n) ∑_{j=[\frac{n+m}{2}]}^{±∞} q(j), ±m > ±n.

Here the functions q_{±}(n) and C_{±}(n) are of the same type as in (4.6).

Moreover, the following estimates are valid

\begin{matrix}∑_{n=n_{0}}^{±∞} |n||K_{±}(n, n) − K_{±}(n ± 1, n ± 1)| < ∞, \\ ∑_{n=n_{0}}^{±∞} |n||a_{q}^{±}(n)K_{±}(n, n + 1) − a_{q}^{±}(n − 1)K_{±}(n − 1, n)| < ∞. \end{matrix} (5.2)

Proof. The solvability of (4.1) under condition (4.6) and the estimates (5.1), (5.2) follow completely analogous to the corresponding result in [8, Theorem 7.5]. To prove uniqueness, first note that the GLM equations are generated by compact operators. Thus, it is sufficient to prove that the equation

(5.3)  f(m) + ∑_{ℓ=n}^{±∞} F_{±}(ℓ, m)f(ℓ) = 0

has only the trivial solution in the space ℓ^{1}(n, ±∞). The proof is similar for the ”+” and ”−” cases, hence we give it only for the ”+” case. Let f(ℓ), ℓ > n, be a nontrivial solution of (5.3) and set f(ℓ) = 0 for ℓ ≤ n. Since F_{+}(ℓ, n) is real-valued, we can assume that f(ℓ) is real-valued. Abbreviate by

(5.4)  f̂(λ) = ∑_{m∈ℤ} ψ_{q}^{+}(λ, m)f(m)

the generalized Fourier transform, generated by the spectral decomposition (2.18) (cf. [22]). Recall that f̂(λ) ∈ L_{loc}^{1}(σ_{+}^{u} ∪ σ_{+}^{l}).

Multiplying (5.3) by f(m), summing over m ∈ ℤ, and applying (2.18), (4.2), (5.4), and condition I, (a), we have

\begin{matrix}2 ∫_{σ_{+}^{u}} |f̂(λ)|^{2}dω_{+}(λ) + 2Re ∫_{σ_{+}^{u}} R_{+}(λ)f̂(λ)^{2}dω_{+}(λ) \\ + ∫_{σ_{−}^{(1),u}} f̂(λ)^{2}|T_{−}(λ)|^{2}dω_{−}(λ) + ∑_{k=1}^{p} γ_{+,k}( ∑_{n∈ℤ} ψ̃_{q}^{+}(λ_{k}, n)f(n))^{2} = 0. \end{matrix} (5.5)

The last two summands in (5.5) are nonnegative since f̂(λ) ∈ ℝ for λ ∈ σ_{−}^{(1)} and ψ̃_{q}^{+}(λ_{k}) ∈ ℝ. We estimate the first two integrands by

|f̂(λ)|^{2} + ReR_{+}(λ)f̂(λ)^{2} ≥ |f̂(λ)|^{2} − |R_{+}(λ)f̂(λ)^{2}| ≥ (1 − |R_{+}(λ)|)|f̂(λ)|^{2}

and drop the last summand in (5.5), thus obtaining

(5.6) 2 ∫_{σ^{(2),u}}(1 − |R_{+}(λ)|)|f̂(λ)|^{2}dω_{+}(λ) + ∫_{σ_{−}^{(1),u}} f̂(λ)^{2}|T_{−}(λ)|^{2}dω_{−}(λ) ≤ 0.

Here we also used that

∫_{σ_{+}^{(1),u}}(1 − |R_{+}(λ)|)|f̂(λ)|^{2}dω_{+}(λ) = 0,

which follows from condition I, (b). Since |R_{+}(λ)| < 1 for λ ∈ int(σ^{(2)}) and ω_{−}(λ) > 0 for λ ∈ int(σ_{−}^{(1)}) we conclude that

f̂(λ) = 0 for λ ∈ σ^{(2)} ∪ σ_{−}^{(1)} = σ_{−}.

The function f̂(z) can be defined by formula (5.4) as a meromorphic function on ℂ \ σ_{+}. By our analysis it is even meromorphic on ℂ \ σ_{+}^{(1)} and vanishes on σ_{−}. Thus f̂(z) and hence also f(m) are equal to zero. □

Next, define the sequences a_{±}, b_{±} by

\begin{aligned}a_{+}(n) & = a_{q}^{+}(n)\frac{K_{+}(n + 1, n + 1)}{K_{+}(n, n)}, \\ a_{−}(n) & = a_{q}^{−}(n)\frac{K_{−}(n, n)}{K_{−}(n + 1, n + 1)}, \\ b_{+}(n) & = b_{q}^{+}(n) + a_{q}^{+}(n)\frac{K_{+}(n, n + 1)}{K_{+}(n, n)} − a_{q}^{+}(n − 1)\frac{K_{+}(n − 1, n)}{K_{+}(n − 1, n − 1)}, \\ b_{−}(n) & = b_{q}^{−}(n) + a_{q}^{−}(n − 1)\frac{K_{−}(n, n − 1)}{K_{−}(n, n)} − a_{q}^{−}(n)\frac{K_{−}(n + 1, n)}{K_{−}(n + 1, n + 1)}, \end{aligned} (5.7)

and note that estimate (5.2) implies

(5.8)  n {|a_{±}(n) − a_{q}^{±}(n)| + |b_{±} − b_{q}^{±}(n)|} ∈ ℓ^{1}(ℤ_{±}).

Lemma 5.2. The functions ψ_{±}(z, n), defined by

(5.9)  ψ_{±}(z, n) = ∑_{m=n}^{±∞} K_{±}(n, m)ψ_{q}^{±}(z, m),

solve the equations

(5.10) a_{±}(n − 1)ψ_{±}(z, n − 1) + b_{±}(n)ψ_{±}(z, n) + a_{±}(n)ψ_{±}(z, n + 1) = zψ_{±}(z, n),

where a_{±}(n), b_{±}(n) are defined by (5.7).

Proof. Consider the two operators^{2} (H_{±}y)(n) = a_{±}(n − 1)y_{±}(n − 1) + b_{±}(n)y_{±}(n) + a_{±}(n)y_{±}(n + 1), n ∈ ℤ.

2We don’t know that H_{±} is limit point at ∓∞ yet, but this will not be used.

Define two discrete integral operators

(K_{±}f) (n) = ∑_{m=n}^{±∞} K_{±}(n, m)f(m).

Then (cf. [8]) the following identity is valid

H_{±}K_{±} = K_{±} H_{q}^{±},

which implies (5.10). □

The remaining problem is to show that a_{+}(n) ≡ a_{−}(n), b_{+}(n) ≡ b_{−}(n) under conditions II and III on the scattering data S.

Theorem 5.3. Let the scattering data S, defined as in (4.7), satisfy conditions I, (a)–(c), II, III, (a), and IV. Then each of the GLM equations (4.1) has unique solutions K_{±}(n, m), satisfying the estimate (5.2). The functions a_{±}(n), b_{±}(n), defined by (5.7), satisfy (5.8).

Under the additional conditions III, (b) and I, (d), these functions coincide, a_{+}(n) ≡ a_{−}(n) = a(n), b_{+}(n) ≡ b_{−}(n) = b(n), and the data S are the scattering data for the Jacobi operator associated with the sequences a(n), b(n).

The proof of Theorem 5.3 takes up the remaining section and is split into several lemmas for the convenience of the reader.

To prove uniqueness of the reconstructed potential we follow the method proposed in [15]. Recall that, by Lemma 2.1 (iii), the functions ψ_{q}^{±}(λ, n) form an orthonormal basis with corresponding generalized Fourier transform. Split the kernel of the GLM equation (4.2) into three summands F_{±}(m, n) = F_{r,±}(m, n) + F_{h,±}(m, n) + F_{d,±}(m, n) and set

(5.11)  G_{±}(n, m) := ∑_{l=n}^{±∞} K_{±}(n, l)F_{r,±}(l, n).

Then one obtains as in [8, Theorem 8.2] that the functions h_{∓}(λ, n), defined by

\begin{aligned}h_{∓}(λ, n) & = \frac{1}{T_{±}(λ)}(\frac{ψ̆_{q}^{±}(λ, n)}{K_{±}(n, n)} + ∑_{m=n∓1}^{∓∞} G_{±}(n, m)ψ̆_{q}^{±}(λ, m) \\ & ∓ ∫_{σ_{∓}^{(1),u}} |T_{∓}(ξ)|^{2}ψ_{±}(ξ, n)\frac{W_{q,n−1}^{±}(ψ_{q}^{±}(ξ), ψ̆_{q}^{±}(λ))}{ξ − λ}dω_{∓}(ξ) \\ & ± ∑_{k=1}^{p} γ_{±,k}ψ̃_{±}(λ_{k}, n)\frac{W_{q,n−1}^{±}(ψ̃_{q}^{±}(λ_{k}), ψ̆_{q}^{±}(λ))}{λ − λ_{k}}), \end{aligned} (5.12)

satisfy

(5.13)  T_{±}(λ)h_{∓}(λ, n) = \overline{ψ_{±}(λ, n)} + R_{±}(λ)ψ_{±}(λ, n), λ ∈ σ_{±}^{u,l}.

Despite the fact that h_{∓}(λ, n) are defined via the background solutions corresponding to the opposite half-axis ℤ_{±}, they share a series of properties with ψ_{∓}(λ, n). Namely, we prove

Lemma 5.4. Let h_{∓}(z, n) be defined by formula (5.12) on the set σ_{±}^{u,l}.

(i) The functions h̃_{∓}(z, n) = δ_{∓}(z)h_{∓}(z, n) admit analytic extensions to the

domain ℂ \ σ.

(ii) The functions h̃_{∓}(z, n) are continuous up to the boundary σ^{u,l} except possibly at the points ∂σ_{+} ∪ ∂σ_{−}. Furthermore,

\begin{matrix}h̃_{∓}(λ^{u}, n) = h̃_{∓}(λ^{l}, n) ∈ ℝ,  λ ∈ ℝ \ σ_{∓}, \\ h̃_{∓}(λ^{u}, n) = \overline{h̃_{∓}(λ^{l}, n)},  λ ∈ int(σ_{∓}). \end{matrix} (5.14)

(iii) For large z the functions h_{∓}(z, n) have the following asymptotic behavior

(5.15) h_{∓}(z, n) = \frac{z^{±n}}{K_{±}(n, n)T_{±}(∞)}( \begin{matrix}n−1 \\ ∏ & ∗ \\ j=0 \end{matrix} a_{q}^{±}(j))^{∓1}(1 + O(\frac{1}{z})), z → ∞.

(iv) We have

W^{±}(h_{∓}(z), ψ_{±}(z)) :=a_{±}(n)(h_{∓}(z, n)ψ_{±}(z, n + 1) − h_{∓}(z, n + 1)ψ_{±}(z, n))
≡ ± W (z),

where W (z) is defined by (3.19).

Remark 5.5. Note that we did not establish the connection between the function W (z) and the functions W^{±}(ψ_{+}(z, n), ψ_{−}(z, n)), which can depend on n, because ψ_{+} and ψ_{−} are the solutions of Jacobi equations corresponding to possibly different operators H_{+} and H_{−}.

Proof. (i). To show that h̃_{∓}(z, n) have analytic extensions to ℂ \ σ, we study each term in (5.12) separately.

First of all, note that due to the representation

(5.16)  T_{±}(z) = \frac{1}{ρ_{±}(z)W (z)} = \frac{δ̂_{∓}(z)}{δ̆_{±}(z)} \frac{\sqrt{∏_{j=0}^{2g_{±}+1}(z − E_{j}^{±})}}{Ŵ (z)},

the functions ζ̃_{∓}(z, n) = δ_{∓}(z)ζ_{∓}(z, n), where

(5.17)  ζ_{∓}(z, n) := \frac{ψ̆_{q}^{±}(z, n)}{T_{±}(z)},

can be continued analytically to ℂ \ σ. This also holds for the second term since G_{±}(n, ·) ∈ ℓ^{1}(ℤ) are real-valued.

Next we discuss the properties of the Cauchy-type integral in the representation (5.12). We represent the third summand in (5.12) multiplied by T_{±}^{−1}(z) as

(5.18)  Θ_{∓}(z, n) := ∓\frac{1}{2πi} ∫_{σ_{∓}^{(1),u}} θ_{∓}(z, ξ, n)\frac{dξ}{ξ − z} ,

where

θ_{∓}(z, ξ, n) = −\frac{δ_{∓}(ξ)^{2}}{ρ_{∓}(ξ)|W̃ (ξ)|^{2}} ψ̃_{±}(ξ, n)W_{q,n−1}^{±}(ψ̃_{q}^{±}(ξ, ·), ζ_{∓}(z, ·))

= −\frac{|δ̂_{∓}(ξ)|^{2}}{ρ_{∓}(ξ)|Ŵ (ξ)|^{2}} \frac{|δ̂_{±}(ξ)|^{2}}{δ̂_{±}(ξ)^{2}} ψ̂_{±}(ξ, n)W_{q,n−1}^{±}(ψ̂_{q}^{±}(ξ, ·), ζ_{∓}(z, ·)). (5.19)

By property II, (a) the function Ŵ (ξ) has no zeros in the interior of σ_{∓}^{(1),u}. Thus, for z ∉ σ_{∓}^{(1)}, the functions θ_{∓}(z, ., n) are bounded in the interior of σ_{∓}^{(1)} and the only possible singularities can arise at the boundary. We claim

(5.20)  θ_{∓}(z, ξ, n) = \begin{cases}O(\sqrt{ξ − E}) & for E ∉ σ_{v}, \\ O(\frac{1}{\sqrt{ξ−E}} ) & for E ∈ σ_{v}, \end{cases}  E ∈ ∂σ_{∓}^{(1)}, z ≠ E.

This follows from \frac{|δ̂_{∓}(ξ)|^{2}}{ρ_{∓}(ξ)} = O(\sqrt{ξ − E}) together with Ŵ (ξ) = O(1) if E ∉ σ_{v} and

1/Ŵ (ξ) = O(1/\sqrt{ξ − E}) by II, (b) if E ∈ σ_{v}. Therefore, θ_{∓} are integrable and the third summand of (5.12) also inherits the properties of ζ_{∓}(z, n).

Finally, the last summand in (5.12) again inherits the properties of ζ̃_{∓}(z, n) except for possible additional poles at the eigenvalues λ_{k}. However, these cancel with the zeros of W̃ (z) at z = λ_{k}.

(ii). We consider the boundary values next. The only nontrivial term is of course the Cauchy-type integral (5.18) as z → λ ∈ int(σ_{∓}^{(1)}). First of all observe that by (2.7) and (3.19),

\frac{W_{q,n−1}^{±}(ψ̃_{q}^{±}(λ), ψ̆_{q}^{±}(z))}{T_{±}(z)} → (δ_{±}W)(λ),

where the functions δ_{±}W are bounded and nonzero for λ ∈ int(σ_{∓}^{(1)}) by II, (a). Hence the Plemelj formula applied to (5.18) gives

Θ_{∓}(λ, n) = ±\frac{ψ̃_{±}(λ, n)}{2δ_{±}(λ)ρ_{∓}(λ)\overline{W (λ)}} ∓ − ∫_{σ_{∓}^{(1),u}} \frac{θ_{∓}(λ, ξ, n)}{ξ − λ}dξ, λ ∈ int(σ_{∓}^{(1),u}),

where both terms are finite. Here ^{−} ∫ denotes the principle value integral. Therefore, the boundary values away from ∂σ_{+} ∪ ∂σ_{−} exist and we have

(5.21)  h_{∓}(λ^{u}, n) = \overline{h_{∓}(λ^{l}, n)}, λ ∈ σ_{+} ∪ σ_{−}.

By property I, (b),

(5.22)  h_{∓} = T_{±}^{−1} (R_{±}ψ_{±} + \overline{ψ_{±}}) = \frac{ψ_{±}}{\overline{T_{±}}} + \frac{\overline{ψ_{±}}}{T_{±}} ∈ ℝ, λ ∈ σ_{±}^{(1)},

from which

(5.23)  h_{∓}(λ^{u}, n) = h_{∓}(λ^{l}, n), λ ∈ σ_{±}^{(1)},

follows. Combining (5.21) and (5.23) yields (5.14).

(iii). Since the last two terms in (5.12) are O(z^{−1}), the asymptotic behavior follows from (3.5) and II, (c).

(iv). From (5.13), (3.8), and (3.19) we obtain

W^{±}(h_{∓}(λ), ψ_{±}(λ)) = \frac{W^{±}(\overline{ψ_{±}(λ)}, ψ_{±}(λ))}{T_{±}(λ)} = \frac{1}{T_{±}(λ)ρ_{±}(λ)} = ±W (λ), λ ∈ σ_{±}.

Hence equality holds for all z ∈ ℂ by analytic continuation. □

Corollary 5.6. The functions h̃_{∓}(z, n) admit analytic extensions to ℂ \ σ_{∓}.

Proof. Property (i) of Lemma 5.4 holds for z ∈ ℂ \ σ. Relation (5.14) implies that h̃_{∓} have no jumps across z ∈ int(σ_{±}^{(1)}). To finish the proof we need to show that the possible remaining singularities at E ∈ ∂σ_{±}^{(1)} ∩ ∂σ are removable. This follows from (cf. (5.16))

(5.24)  ζ̂_{∓}(z, n) = \frac{Ŵ (z)}{\sqrt{∏_{j=0}^{2g_{±}+1}(z − E_{j}^{±})}}δ̆_{±}(z)ψ̆_{q}^{±}(z, n)

which shows ζ̃_{∓}(z, n) = O((z − E)^{−1/2}) and hence h̃_{∓}(z, n) = O((z − E)^{−1/2}) for E ∈ σ_{±}^{(1)} ∩ ∂σ.

However, let us emphasize at this point that the behavior of h_{±}(z, n) at the remaining edges is a more subtle question to be discussed later. □

Eliminating \overline{ψ_{±}} from

\{ \begin{matrix}\overline{R_{±}(λ)} \overline{ψ_{±}(λ, n)} + ψ_{±}(λ, n) & = & \overline{h_{∓}(λ, n)} \overline{T_{±}(λ)} \\ R_{±}(λ) ψ_{±}(λ, n) + \overline{ψ_{±}(λ, n)} & = & h_{∓}(λ, n) T_{±}(λ) \end{matrix}

yields

ψ_{±}(λ, n) (1 − |R_{±}(λ)|^{2}) = \overline{h_{∓}(λ, n)} \overline{T_{±}(λ)} − \overline{R_{±}(λ)} h_{∓}(λ, n) T_{±}(λ).

We apply I, (c), II, and the consistency condition I, (d) to obtain

T_{∓}(λ)ψ_{±}(λ, n) = \overline{h_{∓}(λ, n)} − \frac{\overline{R_{±}(λ)}T_{±}(λ)}{\overline{T_{±}(λ)}}h_{∓}(λ, n)

= \overline{h_{∓}(λ, n)} + R_{∓}(λ)h_{∓}(λ, n), λ ∈ σ^{(2)}. (5.25)

This equation together with (5.13) gives us a system from which we can eliminate the reflection coefficients R_{±}. We obtain (5.26) T_{±}(λ)(ψ_{±}(λ)ψ_{∓}(λ) − h_{±}(λ)h_{∓}(λ)) = ψ_{±}(λ)\overline{h_{±}(λ)} − \overline{ψ_{±}(λ)}h_{±}(λ), λ ∈ σ^{(2),u,l}.

Now introduce the function

(5.27)  G(z) := G(z, n) = \frac{ψ_{+}(z, n)ψ_{−}(z, n) − h_{+}(z, n)h_{−}(z, n)}{W (z)}

which is well defined in the domain z ∈ ℂ \ (σ ∪ σ_{d} ∪ M_{+} ∪ M_{−}). By (5.26) and (3.19),

(5.28)  G(λ) = (ψ_{±}(λ)\overline{h_{±}(λ)} − \overline{ψ_{±}(λ)}h_{±}(λ)) ρ_{±}(λ), λ ∈ σ^{(2),u,l},

so we need to study the properties of G(z, n) as a function of z. Our aim is to prove that G(z, n) = 0, which will follow from the next lemma.

Lemma 5.7. The function G(z, n), defined by (5.27), has the following properties. (i) G(λ^{u}, n) = G(λ^{l}, n) ∈ ℝ for λ ∈ ℝ \ (∂σ_{−} ∪ ∂σ_{+} ∪ σ_{d}). (ii) It has removable singularities at the points ∂σ_{−} ∪ ∂σ_{+} ∪ σ_{d}, where σ_{d} :=

{λ_{1}, ..., λ_{p}}.

Proof. (i). We can rewrite G(z, n) as

(5.29)  G(z, n) = \frac{ψ̃_{+}(z, n)ψ̃_{−}(z, n) − h̃_{+}(z, n)h̃_{−}(z, n)}{W̃ (z)},

where h̃_{±}(z, n) = δ_{±}(z)h_{±}(z, n) as usual. The numerator is bounded near the points under consideration and the denominator does not vanish there. Thus G(z, n) has no singularities at the points (M_{+} ∪ M_{−}) \ σ_{d}.

Furthermore, by Lemma 5.4, II, (a), and Lemma 3.1 we know that G(z, n) has continuous limiting values on the sets σ_{−} and σ_{+}, except possibly at the edges, and satisfies

G(λ^{u}, n) = \overline{G(λ^{l}, n)}, λ ∈ σ_{+} ∪ σ_{−}.

Hence, if we can show that these limits are real, they will be equal and G(z, n) will extend to a meromorphic function on ℂ, that is, (i) holds. To this aim we first observe that (5.14), (5.28), and Lemma 3.1 imply

(5.30)  G(λ^{u}, n) = G(λ^{l}, n) ∈ ℝ, λ ∈ int(σ^{(2)}).

Thus, it remains to prove

(5.31)  G(λ^{u}, n) = G(λ^{l}, n) ∈ ℝ for λ ∈ int(σ_{−}^{(1)}) ∪ int(σ_{+}^{(1)}).

Let us show that G(λ, n) has no jump on the set int(σ_{−}^{(1)})∪int(σ_{+}^{(1)}). We abbreviate

(5.32)  [G] := G(λ) − \overline{G(λ)} = [\frac{ψ_{+}ψ_{−}}{W} ] − [\frac{h_{+}h_{−}}{W} ] , λ ∈ σ_{±}^{(1),u},

and drop some dependencies until the end of this lemma for notational simplicity.

Let λ ∈ int(σ_{∓}^{(1),u}), then ψ_{±}, h_{±} ∈ ℝ and \overline{T}_{∓} = −(\overline{W} ρ_{∓})^{−1}. By (3.19), (I), (b), and (5.13) we obtain for λ ∈ int(σ_{∓}^{(1)})

(5.33)  [\frac{ψ_{+}ψ_{−}}{W} ] = ψ_{±} [\frac{ψ_{∓}}{W} ] = ρ_{∓}ψ_{±} (ψ_{∓}T_{∓} + \overline{ψ}_{∓}\overline{T}_{∓}) = ρ_{∓}h_{±}ψ_{±}|T_{∓}|^{2}.

Since ρ_{±} ∈ ℝ for λ ∈ int(σ_{∓}^{(1),u}), (3.19) implies

[\frac{h_{∓}}{W} ] = ρ_{±} [h_{∓}T_{±}] .

The only non-real summand in (5.12) is the Cauchy-type integral. The Plemelj formula applied to this integral gives

[h_{∓}T_{±}] = −ρ_{∓}ψ_{±}|T_{∓}|^{2}W (ψ_{q}^{±}, ψ̆_{q}^{±}) = ρ_{∓}ψ_{±}|T_{∓}|^{2} \frac{1}{ρ_{±}} ,

and by (5.33) we get

(5.34)  [\frac{h_{+}h_{−}}{W} ] = [\frac{ψ_{+}ψ_{−}}{W} ] = ρ_{∓}ψ_{±}h_{±}|T_{∓}|^{2}, λ ∈ int(σ_{∓}^{(1)}).

Since W̃ ≠ 0 for λ ∈ int(σ_{∓}^{(1)}), the function

ρ_{∓}ψ_{±}h_{±}|T_{∓}|^{2} = −\frac{δ_{∓}^{2}}{ρ_{∓}} \frac{ψ̃_{±}h̃_{±}}{|W̃ |^{2}}

is bounded on the set under consideration. Finally, (5.34) and (5.32) imply (5.31).

(ii). Now we prove that the function G(z, n) has removable singularities at the points ∂σ_{−} ∪ ∂σ_{+} ∪ σ_{d}. We divide this set into four subsets

(5.35) Ω_{1}^{±} = ∂σ^{(2)} ∩ int(σ_{∓}), Ω_{2} = ∂σ^{(2)} ∩ ∂σ, Ω_{3}^{±} = ∂σ_{±}^{(1)} ∩ ∂σ_{±}, Ω_{4} = σ_{d}.

Since all singularities of G are at most isolated poles, it is sufficient to show that

(5.36)  G(z) = o((z − E)^{−1})

from some direction in the complex plane.

Ω_{1}: Consider E ∈ Ω_{1}^{+} (the case E ∈ Ω_{1}^{−} being completely analogous). We will study lim_{λ→E} G(λ, n) as λ ∈ int(σ^{(2)}) using (5.28) with the “−” sign. Note that ψ_{−} = O(1), ρ_{−} = O(1), and Ŵ (E) ≠ 0. Moreover, we obtain from Lemma 3.1 respectively II that

ψ_{+}(λ) = \{ \begin{matrix}O(1), & E ∉ M̂_{+}, \\ O (\frac{1}{\sqrt{λ−E}} ) , & E ∈ M̂_{+}, \end{matrix} \frac{1}{T_{+}(λ)} = \{ \begin{matrix}O (\frac{1}{\sqrt{λ−E}} ) , & E ∉ M̂_{+}, \\ O(1), & E ∈ M̂_{+}, \end{matrix}

which shows

h_{−}(λ) = \frac{\overline{ψ_{+}(λ)} + R_{+}(λ)ψ_{+}(λ)}{T_{+}(λ)} = O (\frac{1}{\sqrt{λ − E}} )

for λ ∈ σ^{(2)}. Inserting this into (5.28) shows G(λ, n) = O(\frac{1}{\sqrt{λ−E}} ) and finishes the case E ∈ Ω_{1}.

Ω_{2}: For E ∈ ∂σ^{(2)} ∩ ∂σ, we use (5.28) and take the limit λ → E from σ^{(2)}. First of all, observe that

δ̆_{−} (R_{−}ψ_{−} + \overline{ψ}_{−}) = \{ \begin{matrix}O(1) & E ∈ σ_{v}, \\ o(1) & E ∉ σ_{v}. \end{matrix}

The case E ∈ σ_{v} is evident. If E ∉ σ_{v} then (3.11) and (3.25) yield

δ̆_{−} (R_{−}ψ_{−} + \overline{ψ}_{−}) = \{ \begin{matrix}δ̆_{−}((ψ_{−} − \overline{ψ}_{−}) + (R_{−} + 1)ψ_{−}), & E ∉ M̂_{−} \\ (δ̆_{−}(ψ_{−} + \overline{ψ}_{−}) + (R_{−} − 1)δ̆_{−}ψ_{−}), & E ∈ M̂_{−} \end{matrix} = o(1).

Therefore, both for virtual and non-virtual levels the estimate

(5.37)  δ̆_{−} (R_{−}ψ_{−} + \overline{ψ}_{−}) Ŵ = o(1), E ∈ ∂σ_{−},

is valid. Inserting (5.13) into the summand \overline{ψ}_{+}h_{+}ρ_{+} of (5.28) (for the second summand we use an analogous approach) we obtain (recall (2.2))

\overline{ψ}_{+}h_{+}ρ_{+} = \overline{ψ}_{+}ρ_{+}ρ_{−}(\overline{ψ}_{−} + R_{−}ψ_{−})W = \frac{\overline{ψ}_{+}δ̆_{+}}{P_{+}P_{−}} δ̂_{+}δ̂_{−}δ̆_{−}(\overline{ψ}_{−} + R_{−}ψ_{−})W

= \frac{\overline{ψ}_{+}δ̆_{+}}{P_{+}P_{−}} δ̆_{−} (R_{−}ψ_{−} + \overline{ψ}_{−}) Ŵ . (5.38)

Combining the estimate

\frac{\overline{ψ}_{+}δ̆_{+}}{P_{+}P_{−}} = O(\frac{1}{λ − E} )

with (5.37) we have G(z) = o((z − E)^{−1}) as desired.

Ω_{3}: Suppose that E ∈ ∂σ_{−}^{(1)} ∩∂σ_{−} (the case E ∈ ∂σ_{+}^{(1)} ∩∂σ_{+} is again analogous). Now we cannot use (5.28), so we proceed directly from formula (5.27) estimating the summands \frac{ψ_{+}ψ_{−}}{W} and \frac{h_{+}h_{−}}{W} separately. We investigate the limit as λ → E from the set int(σ_{−}^{(1)}). By Lemma 3.1 and (3.22) we have

(5.39)  \frac{ψ_{+}ψ_{−}}{W} = \frac{ψ̂_{+}ψ̂_{−}}{Ŵ} = O (\frac{1}{\sqrt{λ − E}} ) ,

hence the first summand has the desired behavior. To estimate the second summand, we split the function h_{−}(λ, n) according to

h_{−}(λ, n) = h_{1}(λ, n) + h_{2}(λ, n),

20 I. EGOROVA, J. MICHOR, AND G. TESCHL

where

h_{1}(λ, n) = W_{q,n−1}^{+}(ζ_{−}(λ, ·), d_{−}(λ, n, ·)), h_{2}(λ, n) = h_{−}(λ, n) − h_{1}(λ, n),
(5.40)  d_{−}(λ, n, .) := ∫_{σ_{−}^{(1),u}} \frac{|T_{−}(ξ)|^{2}ψ_{+}(ξ, n)ψ_{q}^{+}(ξ, ·)}{ξ − λ}dω_{−}(ξ).

It follows from the proof of Lemma 5.4 that h_{2}(λ) = O(ζ_{−}(λ)) for λ → E. Recall that at the point under consideration singularities E ∈ {μ_{1}^{+}, …, μ_{g_{+}}^{+}} ∪ M̂_{−} might occur (in the case ∂σ_{−}^{(1)}∩∂σ one can have E ∈ M_{+}∪M̆_{+} and in the case ∂σ_{−}^{(1)}∩∂σ_{+}^{(1)} one can have E ∈ M̂_{+}). Introduce

(5.41)  ϕ_{q}^{+}(z, n) := δ̆_{+}(z)ψ̆_{q}^{+}(z, n)

and recall that (2.15) implies

(5.42)  ϕ_{q}^{+}(z, n) − ϕ_{q}^{+}(E, n) = O(\sqrt{z − E}).

Then (see (2.2) and (2.13)) we have

(5.43)  \frac{h_{+}ζ_{−}}{W} = O(\frac{h_{+}ψ̆_{q}^{+}}{W T_{+}} ) = O(\frac{h_{+}δ̂_{+}δ̆_{+}ψ̆_{q}^{+}}{P_{+}} ) = O(\frac{h_{+}δ̂_{+}}{P_{+}} )ϕ_{q}^{+}.

Now we distinguish two cases: (a) E ∈ ∂σ_{−}^{(1)} ∩ ∂σ_{+}^{(1)} and (b) E ∈ ∂σ_{−}^{(1)} ∩ ∂σ.

Case (a). By (5.13) and (5.37) we have

(5.44) δ̂_{+}h_{+} = \frac{(R_{−}ψ_{−} + \overline{ψ}_{−})δ̂_{+}}{T_{−}} = \frac{Ŵ δ̆_{−}(R_{−}ψ_{−} + \overline{ψ}_{−})}{P_{−}} = o(\frac{1}{\sqrt{λ − E}} ),

therefore

(5.45)  \frac{h_{+}(λ)ζ_{−}(λ)}{W (λ)} = o(\frac{1}{\sqrt{λ − E}} )\frac{ϕ_{q}^{+}(λ)}{P_{+}(λ)}.

As a consequence of \frac{ϕ_{q}^{+}}{P_{+}} = O(\frac{1}{\sqrt{λ−E}} ) we obtain

(5.46)  \frac{h_{+}h_{2}}{W} = o(\frac{1}{λ − E} ), E ∈ ∂σ_{−}.

Next, we have to estimate

(5.47)  \frac{h_{+}h_{1}}{W} = W_{q,n−1}^{+}(\frac{h_{+}ζ_{−}}{W} , d_{−}).

By (5.42) we can represent (5.45) as

(5.48)  \frac{h_{+}(λ)ζ_{−}(λ)}{W (λ)} = o(\frac{1}{\sqrt{λ − E}} )(\frac{ψ̄_{q}^{+}(E)}{\sqrt{λ − E}} + O(1)).

Then (5.47) implies

\begin{aligned}\frac{h_{+}(λ, n)h_{1}(λ, n)}{W (λ)} & = o(\frac{1}{\sqrt{λ − E}} )(O(d_{−}(λ, n)) + O(d_{−}(λ, n − 1)) \\ & + \frac{O (W_{q,n−1}^{+} (ϕ_{q}^{+}(E), d_{−}(λ)))}{\sqrt{λ − E}}). \end{aligned} (5.49)

To estimate d_{−} in the first two summands we distinguish between the resonance case, E ∈ σ_{v}, and non-resonance, E ∉ σ_{v}. First let E ∉ σ_{v}, that is, Ŵ (E) ≠ 0.

From (5.19) and (5.20) we see that the integrand is bounded as λ → E ∉ σ_{v}, then d_{−}(λ) = O(1) by [17].

If E ∈ σ_{v}, then (3.22) (see also (5.19)) yields

|T_{−}(ξ)|^{2}ρ_{−}(ξ)ψ_{+}(ξ, ·)ψ_{q}^{+}(ξ, ·) = O (\frac{1}{\sqrt{ξ − E}} )

and [17, Eq. (29.8)] implies

(5.50)  d_{−}(λ) = o (\frac{1}{\sqrt{λ − E}} ) .

For the estimate of the last summand in (5.49) we use (5.19) and (5.40) to represent the integrand in W_{q,n−1}^{+} (ψ̆_{q}^{+}(E), d_{−}(λ)) as

|T_{−}(ξ)|^{2}ρ_{−}(ξ)ψ_{+}(ξ, n)W_{q,n−1}^{+} (ψ_{q}^{+}(ξ), ϕ_{q}^{+}(E))

= O(\frac{\sqrt{ξ − E}}{|Ŵ (ξ)|^{2}} W_{q,n−1}^{+}(ψ̂_{q}^{+}(ξ), ϕ_{q}^{+}(E))).

It follows from (2.15) and (5.41) that

W_{q,n−1}^{+}(ψ̂_{q}^{+}(ξ), ϕ_{q}^{+}(E)) = O(\sqrt{ξ − E}),

which implies together with (3.22) the boundedness of the integrand near E. Thus,

(5.51)  W_{q,n−1}^{+} (ϕ_{q}^{+}(E), d_{−}(λ)) = O(1),

and combining (5.46), (5.49), (5.51), and (5.50) finishes case (a).

Case (b). Now we do not have estimate (5.37) (cf. III, (b)) at our disposal, but we can proceed as in (5.43), (5.44) since P_{+}(E) ≠ 0 and arrive at

(5.52)  \frac{h_{+}ζ_{−}}{W} = O(h_{+}δ̂_{+}) = O(\frac{Ŵ δ̆_{−}(R_{−}ψ_{−} + \overline{ψ}_{−})}{P_{−}}) = O(\frac{Ŵ}{\sqrt{λ − E}} ).

This estimate is sufficient to conclude that (5.46) is valid in case (b) as well. For h_{1}, we use the following estimate (cf. (5.50) and (5.52)) instead of (5.47):

\frac{h_{+}h_{1}}{W} = O (\frac{h_{+}ζ_{−}}{W} ) O (d_{−}) = O (\frac{Ŵ}{\sqrt{λ − E}} ) o (\frac{1}{\sqrt{λ − E}} ) .

Combining this with (5.46) finishes case (b).

Ω_{4}: Finally we have to show that the singularities of G(z, n) at the points of the discrete spectrum are removable. Since W̃ (z) has simple zeros at z = λ_{k}, it suffices by (5.29) to show that

(5.53)  h̃_{+}(λ_{k}, n)h̃_{−}(λ_{k}, n) = ψ̃_{−}(λ_{k}, n)ψ̃_{+}(λ_{k}, n).

By Lemma 5.4, the functions h̃_{∓} = δ_{∓}h_{∓} given in (5.12) are continuous at the points M̆_{±}. Since (δ_{∓}T_{±}^{−1})(λ_{k}) = 0 and (δ_{∓}T_{±}^{−1}ψ̆_{q}^{±})(λ_{k}) = 0, only the last summand in (5.12) is non-zero. We compute the limit of this summand as λ → λ_{k} using (3.19),

(5.54)  h̃_{∓}(λ_{k}) = −γ_{±,k}ψ̃_{±}(λ_{k})\frac{dW̃ (λ_{k})}{dλ},

and apply (3.20) to obtain (5.53). □

The identity G(z, n) ≡ 0 implies

(5.55)  ψ_{+}(z, n)ψ_{−}(z, n) − h_{+}(z, n)h_{−}(z, n) ≡ 0, ∀n ∈ ℤ.

For z → ∞ we obtain by (2.16) and (5.9)

ψ_{+}(z, n)ψ_{−}(z, n) = K_{+}(n, n)K_{−}(n, n) \begin{matrix}n−1 \\ ∏ & ∗ \\ j=0 \end{matrix} \frac{a_{q}^{+}(j)}{a_{q}^{−}(j)}(1 + o(1)).

Formulas (5.15) and (3.24) imply

h_{+}(z, n)h_{−}(z, n) = \frac{1}{T_{+}(∞)^{2}K_{+}(n, n)K_{−}(n, n)} \begin{matrix}n−1 \\ ∏ & ∗ \\ j=0 \end{matrix} \frac{a_{q}^{−}(j)}{a_{q}^{+}(j)}(1 + o(1))

and by (5.55),

K_{+}(n, n)K_{−}(n, n) \begin{matrix}n−1 \\ ∏ & ∗ \\ j=0 \end{matrix} \frac{a_{q}^{+}(j)}{a_{q}^{−}(j)} = \frac{1}{T_{+}(∞)}.

The value on the left hand side does not depend on n, so using (5.7) we conclude

(5.56)  a_{+}(n) = a_{−}(n) ≡ a(n), ∀n ∈ ℤ.

It remains to prove b_{+}(n) = b_{−}(n). If we eliminate the reflection coefficient R_{±} from (5.13) at n and (5.25) at n + 1 we obtain

\begin{matrix}G_{1}(λ, n) := \frac{ψ_{+}(λ, n)ψ_{−}(λ, n + 1) − h_{+}(λ, n + 1)h_{−}(λ, n)}{W (λ)} \\ = ρ_{+}(λ)(\overline{h_{±}(λ, n + 1)}ψ_{±}(λ, n) − \overline{ψ_{±}(λ, n)}h_{±}(λ, n + 1)), λ ∈ σ^{(2),u,l}. \end{matrix} (5.57)

Proceeding as for G(λ, n) in Lemma 5.7 we can show that that the function G_{1}(z, n) is holomorphic in ℂ. From (5.15), (5.9), (3.24), (2.16), (5.56), and the Liouville theorem we conclude that

\frac{ψ_{+}(z, n)ψ_{−}(z, n + 1) − h_{+}(z, n + 1)h_{−}(z, n)}{W (z)} = −1/a(n).

We compute the asymptotics of

W̄ (z, n) := a(n) (ψ_{+}(z, n)ψ_{−}(z, n + 1) − h_{+}(z, n + 1)h_{−}(z, n)) = −W (z)

as z → ∞ and obtain (compare (3.5))

(5.58)  0 = W̄ (z, n) − W̄ (z, n − 1) = (b_{+}(n) − b_{−}(n))K_{+}(0, 0)K_{−}(0, 0).

This implies in particular b_{+}(n) = b_{−}(n) ≡ b(n), hence the proof of Theorem 5.3 is finished.

Acknowledgments. I.E. and J.M. gratefully acknowledge the extraordinary hospitality of the Faculty of Mathematics of the University of Vienna during their stay in 2007, where parts of this paper were written.

References

[1] J. Bazargan and I. Egorova, Jacobi operator with step-like asymptotically periodic coefficients, Mat. Fiz. Anal. Geom. 10, No.3, 425–442 (2003).

[2] A. Boutet de Monvel and I. Egorova, The Toda lattice with step-like initial data. Soliton asymptotics, Inverse Problems 16, No.4, 955–977 (2000).

[3] A. Boutet de Monvel, I. Egorova, and E. Khruslov, Soliton asymptotics of the Cauchy problem solution for the Toda lattice, Inverse Problems 13, No.2, 223–237 (1997).

[4] A. Boutet de Monvel, I. Egorova, and G. Teschl, Inverse Scattering theory for one-dimensional Schrödinger operators with steplike periodic potentials, Preprint, arXiv:0707.4632.

[5] K. M. Case, Orthogonal polynomials from the viewpoint of scattering theory, J. Math. Phys. 14, 2166–2175 (1973).

[6] P. Deift, S. Kamvissis, T. Kriecherbauer, and X. Zhou, The Toda rarefaction problem, Comm. Pure Appl. Math. 49, No.1, 35–83 (1996).

[7] I. Egorova, The scattering problem for step-like Jacobi operator, Mat. Fiz. Anal. Geom. 9, No.2, 188–205 (2002).

[8] I. Egorova, J. Michor, and G. Teschl, Scattering theory for Jacobi operators with quasiperiodic background, Comm. Math. Phys. 264-3, 811–842 (2006).

[9] I. Egorova, J. Michor, and G. Teschl, Inverse scattering transform for the Toda hierarchy with quasi-periodic background, Proc. Amer. Math. Soc. 135, 1817–1827 (2007).

[10] I. Egorova, J. Michor, and G. Teschl, Soliton solutions of the Toda hierarchy on quasiperiodic background revisited, Math. Nach. (to appear).

[11] I. Egorova, J. Michor, and G. Teschl, Scattering theory for Jacobi operators with steplike quasi-periodic background, Inverse Problems 23, 905–918 (2007).

[12] G. S. Guseinov, The inverse problem of scattering theory for a second-order difference equation on the whole axis, Soviet Math. Dokl. 17, 1684–1688 (1976).

[13] S. Kamvissis and G. Teschl, Stability of periodic soliton equations under short range perturbations, Phys. Lett. A, 364-6, 480–483 (2007).

[14] S. Kamvissis and G. Teschl, Stability of the periodic Toda lattice under short range perturbations, Preprint, arXiv:0705.0346.

[15] V. A. Marchenko, Sturm–Liouville Operators and Applications, Birkhäuser, Basel, 1986.

[16] J. Michor and G. Teschl, Trace formulas for Jacobi operators in connection with scattering theory for quasi-periodic background, Operator Theory, Analysis, and Mathematical Physics, J. Janas, et al. (eds.), 69–76, Oper. Theory Adv. Appl., 174, Birkhäuser, Basel, 2007.

[17] N. I. Muskhelishvili, Singular integral equations, P. Noordhoff Ltd., Groningen, 1953.

[18] G. Teschl, Oscillation theory and renormalized oscillation theory for Jacobi operators, J. Diff. Eqs. 129, 532–558 (1996).

[19] G. Teschl, On the initial value problem for the Toda and Kac-van Moerbeke hierarchies, in Differential Equations and Mathematical Physics, R. Weikard and G. Weinstein (eds.), 375–384, AMS/IP Studies in Advanced Mathematics 16, Amer. Math. Soc., Providence, 2000.

[20] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surv. and Mon. 72, Amer. Math. Soc., Rhode Island, 2000.

[21] G. Teschl, Algebro-geometric constraints on solitons with respect to quasi-periodic backgrounds, Bull. London Math. Soc. 39, No.4, 677–684 (2007).

[22] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations, Vol. 2, Oxford, at the Clarendon Press, 1958.

[23] S. Venakides, P. Deift, and R. Oba, The Toda shock problem, Comm. Pure Appl. Math. 44, No.8-9, 1171–1242 (1991).

[24] A. Volberg and P. Yuditskii, On the inverse scattering problem for Jacobi Matrices with the Spectrum on an Interval, a finite system of intervals or a Cantor set of positive length, Comm. Math. Phys. 226, 567–605 (2002).

Institute for Low Temperature Physics, 47,Lenin ave, 61164 Kharkiv, Ukraine

E-mail address: egorova@ilt.kharkov.ua

Imperial College, 180 Queen’s Gate, London SW7 2BZ, and International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria

E-mail address: Johanna.Michor@esi.ac.at
URL: http://www.mat.univie.ac.at/~jmichor/

Faculty of Mathematics, Nordbergstrasse 15, 1090 Wien, Austria, and International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria

E-mail address: Gerald.Teschl@univie.ac.at
URL: http://www.mat.univie.ac.at/~gerald/