SOLITON SOLUTIONS OF THE TODA HIERARCHY ON QUASI-PERIODIC BACKGROUNDS REVISITED

IRYNA EGOROVA, JOHANNA MICHOR, AND GERALD TESCHL

Abstract. We investigate soliton solutions of the Toda hierarchy on a quasiperiodic finite-gap background by means of the double commutation method and the inverse scattering transform. In particular, we compute the phase shift caused by a soliton on a quasi-periodic finite-gap background. Furthermore, we consider short-range perturbations via scattering theory. We give a full description of the effect of the double commutation method on the scattering data and establish the inverse scattering transform in this setting.

1. Introduction

Solitons on a (quasi-)periodic background have a long tradition and are used to model localized excitements on a phonon, lattice, or magnetic field background (see, e.g., [5], [11], [13], [15], [16], [17], [18] and the references therein). Of course periodic solutions, as well as solitons travelling on a periodic background, are well understood. Nevertheless there are still several open questions.

One of them is the stability of (quasi-)periodic solutions. For the constant solution it is a classical result, that a small initial perturbation asymptotically splits in a number of stable solitons. For a (quasi-)periodic background this cannot be the case. In fact, associated with every soliton there is a phase shift (which will be explicitly computed in Section 4) and the phase shifts of all solitons will not add up to zero in general. Hence there must be something which makes up for this phase shift. Moreover, even if no solitons are present, the asymptotic limit is not the (quasi-)periodic background! A precise description of the asymptotic limit in terms of Abelian integrals on the underlying Riemann surface is given in [9] (see [10] for a proof). In particular, the asymptotic limit can be split into parts, one which stems from the discrete spectrum (solitons) and one which stems from the continuous spectrum.

The soliton part can be understood by adding/removing the solitons using a Darboux-type transformation, that is, commutation methods for the underlying Jacobi operators. Hence the purpose of the present paper is to complement [10] and provide a detailed description of the double commutation method when applied to a short-range perturbation of a quasi-periodic finite-gap solution of the Toda lattice. In particular, we are interested in the effect of one double commutation step on the scattering data.

2000 Mathematics Subject Classification. Primary 37K15, 37K10; Secondary 47B36, 34L25.

Key words and phrases. Solitons, Toda hierarchy, periodic, inverse scattering transform.

Math. Nachr. 282:4, 526–539 (2009).

Work supported by the Austrian Science Fund (FWF) under Grant No. P17762 and INTAS Research Network NeCCA 03-51-6637.

After introducing the Toda hierarchy in Section 2 and recalling some necessary facts on algebro-geometric quasi-periodic finite-gap solutions in Section 3 we briefly review the single and double commutation methods in Section 4 and compute the phase shift (in the Jacobian variety) caused by inserting one eigenvalue for both methods. In Section 5 we review direct scattering theory for Jacobi operators with different (quasi-)periodic asymptotics in the same isospectral class. As our main result we give a complete description of the effect of the double commutation method on the scattering data. In addition, we provide some detailed asymptotic formulas for the Jost functions ψ_{±}(z, n) (which are normalized as n → ±∞) at the other side, that is, as n → ∓∞. Our final Section 6 establishes the inverse scattering transform for this setting. Our main results here are the time dependence of both the scattering data and the kernel of the Gelfand-Levitan-Marchenko equation.

2. The Toda hierarchy

In this section we introduce the Toda hierarchy using the standard Lax formalism ([12]). We first review some basic facts from [1] (see also [21]).

We will only consider bounded solutions and hence require

Hypothesis H.2.1. Suppose a(t), b(t) satisfy

a(t) ∈ ℓ^{∞}(ℤ, ℝ),  b(t) ∈ ℓ^{∞}(ℤ, ℝ),  a(n, t) ≠ 0,  (n, t) ∈ ℤ × ℝ,

and let t ↦ (a(t), b(t)) be differentiable in ℓ^{∞}(ℤ) ⊕ ℓ^{∞}(ℤ).

Associated with a(t), b(t) is a Jacobi operator

(2.1)  H(t) : ℓ^{2}(ℤ) → ℓ^{2}(ℤ),  f ↦ τ(t)f,

where

(2.2)  τ(t)f(n) = a(n, t)f(n + 1) + a(n − 1, t)f(n − 1) + b(n, t)f(n)

and ℓ^{2}(ℤ) denotes the Hilbert space of square summable (complex-valued) sequences over ℤ. Moreover, choose constants c_{0} = 1, c_{j}, 1 ≤ j ≤ r, c_{r+1} = 0, set

g_{j}(n, t) = ∑_{ℓ=0}^{j} c_{j−ℓ}⟨δ_{n}, H(t)^{ℓ}δ_{n}⟩,

(2.3)  h_{j}(n, t) = 2a(n, t) ∑_{ℓ=0}^{j} c_{j−ℓ}⟨δ_{n+1}, H(t)^{ℓ}δ_{n}⟩ + c_{j+1},

and consider the Lax operator

(2.4)  P_{2r+2}(t) = −H(t)^{r+1} + ∑_{j=0}^{r}(2a(t)g_{j}(t)S^{+} − h_{j}(t))H(t)^{r−j} + g_{r+1}(t),

where S^{±}f(n) = f(n ± 1). Restricting to the two-dimensional nullspace Ker(τ(t) − z), z ∈ ℂ, of τ(t) − z, we have the following representation of P_{2r+2}(t)

(2.5)  P_{2r+2}(t)|_{Ker(τ(t)−z)} = 2a(t)G_{r}(z, t)S^{+} − H_{r+1}(z, t),

where G_{r}(z, n, t) and H_{r+1}(z, n, t) are monic polynomials in z of the type

G_{r}(z, n, t) = ∑_{j=0}^{r} z^{j}g_{r−j}(n, t),

(2.6)  H_{r+1}(z, n, t) = z^{r+1} + ∑_{j=0}^{r} z^{j}h_{r−j}(n, t) − g_{r+1}(n, t).

A straightforward computation shows that the Lax equation

(2.7)  \frac{d}{dt}H(t) − [P_{2r+2}(t), H(t)] = 0,  t ∈ ℝ,

is equivalent to

TL_{r}(a(t), b(t))_{1} = ȧ(t) − a(t)(g_{r+1}^{+}(t) − g_{r+1}(t)) = 0,
(2.8)  TL_{r}(a(t), b(t))_{2} = ḃ(t) − (h_{r+1}(t) − h_{r+1}^{−}(t)) = 0,

where the dot denotes a derivative with respect to t and f^{±}(n) = f(n±1). Varying r ∈ ℕ_{0} yields the Toda hierarchy TL_{r}(a, b) = (TL_{r}(a, b)_{1}, TL_{r}(a, b)_{2}) = 0. We will always consider r as a fixed, but arbitrary, value.

We recall that the Lax equation (2.7) implies existence of a unitary propagator U_{r}(t, s) such that the family of operators H(t), t ∈ ℝ, are unitarily equivalent, H(t) = U_{r}(t, s)H(s)U_{r}(s, t). This also implies the basic existence and uniqueness theorem for the Toda hierarchy (see, e.g., [20], [19], or [21, Section 12.2]).

Theorem 2.2. Suppose (a_{0}, b_{0}) ∈ M = ℓ^{∞}(ℤ) ⊕ ℓ^{∞}(ℤ). Then there exists a unique integral curve t ↦ (a(t), b(t)) in C^{∞}(ℝ, M) of the Toda hierarchy, that is, TL_{r}(a(t), b(t)) = 0, such that (a(0), b(0)) = (a_{0}, b_{0}).

Finally, we recall the following result from [3] (compare also [20]), which says that solutions which are asymptotically close to a background solution at the initial time stay close for all time.

Lemma 2.3. Suppose a(n, t), b(n, t) and ā(n, t), b̄(n, t) are two arbitrary bounded solutions of the Toda hierarchy satisfying (2.9) for one t_{0} ∈ ℝ, then (2.9) holds for all t ∈ ℝ, that is,

(2.9)  ∑_{n∈ℤ} w(n)(|a(n, t) − ā(n, t)| + |b(n, t) − b̄(n, t)|) < ∞,

where w(n) ≥ 1 is some weight with sup_{n}(|\frac{w(n+1)}{w(n)}| + |\frac{w(n)}{w(n+1)}|) < ∞.

3. Quasi-periodic finite-gap solutions

As a preparation for our next section we first need to recall some facts on quasiperiodic finite-gap solutions (again see [1] or [21]).

Let 𝕄 be the Riemann surface associated with the following function

(3.1)  R_{2g+2}^{1/2}(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). 𝕄 is a compact, hyperelliptic Riemann surface of genus g. A point on 𝕄 is denoted by p = (z, ±R_{2g+2}^{1/2}(z)) = (z, ±), z ∈ ℂ, or p = ∞_{±}, and the projection onto ℂ ∪ {∞} by π(p) = z. The sets Π_{±} = {(z, ±R_{2g+2}^{1/2}(z)) | z ∈ ℂ\ ⋃_{j=0}^{g}[E_{2j}, E_{2j+1}]} ⊂ 𝕄 are called upper, lower sheet, respectively.

Now pick g numbers (the Dirichlet eigenvalues)

(3.2)  (μ̂_{j})_{j=1}^{g} = (μ_{j}, σ_{j})_{j=1}^{g}

whose projections lie in the spectral gaps, that is, μ_{j} ∈ [E_{2j−1}, E_{2j}]. Associated with these numbers is the divisor D\underline{μ̂} which is one at the points μ̂_{j} and zero else. Using this divisor we introduce

\underline{z}(p, n, t) = \underline{Â}_{p_{0}}(p) − \underline{α̂}_{p_{0}}(D\underline{μ̂}) − n\underline{Â}_{∞_{−}}(∞_{+}) + t\underline{U}_{s} − \underline{Ξ̂}_{p_{0}} ∈ ℂ^{g},
(3.3)  \underline{z}(n, t) = \underline{z}(∞_{+}, n, t),

where \underline{Ξ}_{p_{0}} is the vector of Riemann constants, \underline{U}_{s} the b-periods of the Abelian differential Ω_{s} defined below, and \underline{A}_{p_{0}} (\underline{α}_{p_{0}}) is Abel’s map (for divisors). The hat indicates that we regard it as a (single-valued) map from 𝕄̂ (the fundamental polygon associated with 𝕄) to ℂ^{g}. We recall that the function θ(\underline{z}(p, n, t)) has precisely g zeros μ̂_{j}(n, t) (with μ̂_{j}(0, 0) = μ̂_{j}), where θ(\underline{z}) is the Riemann theta function of 𝕄.

Taking a stationary solution of TL_{g} with constants c_{j}, 1 ≤ j ≤ g, as initial condition for another equation T̂L_{s} with constants ĉ_{j}, 1 ≤ j ≤ s, in the Toda hierarchy (2.8) one obtains the quasi-periodic finite gap solutions of the Toda hierarchy given by (see [21, Sections 13.1, 13.2])

a_{q}(n, t)^{2} = ã^{2} \frac{θ(\underline{z}(n + 1, t))θ(\underline{z}(n − 1, t))}{θ(\underline{z}(n, t))^{2}},

(3.4)  b_{q}(n, t) = b̃ + ∑_{j=1}^{g} c_{j}(g)\frac{∂}{∂w_{j}} ln (\frac{θ(\underline{w} + \underline{z}(n, t))}{θ(\underline{w} + \underline{z}(n − 1, t))})|_{\underline{w}=0}.

The constants ã, b̃, c_{j}(g) depend only on the Riemann surface (see [21, Section 9.2]). Introduce

ϕ_{q}(p, n, t) = C(n, t)\frac{θ(\underline{z}(p, n + 1, t))}{θ(\underline{z}(p, n, t))} exp ( ∫_{p_{0}}^{p} ω_{∞_{+},∞_{−}}),
(3.5)  ψ_{q}(p, n, t) = C(n, 0, t)\frac{θ(\underline{z}(p, n, t))}{θ(\underline{z}(p, 0, 0))} exp (n ∫_{p_{0}}^{p} ω_{∞_{+},∞_{−}} + t ∫_{p_{0}}^{p} Ω_{s}),

where C(n, t), C(n, 0, t) are real-valued,

(3.6)  C(n, t)^{2} = \frac{θ(\underline{z}(n − 1, t))}{θ(\underline{z}(n + 1, t))},  C(n, 0, t)^{2} = \frac{θ(\underline{z}(0, 0))θ(\underline{z}(−1, 0))}{θ(\underline{z}(n, t))θ(\underline{z}(n − 1, t))},

and the sign of C(n, t) is opposite to that of a_{q}(n, t). ω_{∞_{+},∞_{−}} is the Abelian differential of the third kind with poles at ∞_{+} respectively ∞_{−} and Ω_{s} is an Abelian differential of the second kind with poles at ∞_{+} respectively ∞_{−} whose Laurent expansion is given by the coefficients (j + 1)ĉ_{s−j} associated with T̂L_{s} (see [21, Sections 13.1, 13.2]). Then

τ_{q}(t)ψ_{q}(p, n, t) = π(p)ψ_{q}(p, n, t),
\frac{d}{dt}ψ_{q}(p, n, t) = 2a_{q}(n, t)Ĝ_{s}(p, n, t)ψ_{q}(p, n + 1, t) − Ĥ_{s+1}(p, n, t)ψ_{q}(p, n, t)

= P̂_{q,2s+2}(t)ψ_{q}(p, n, t), (3.7) where we use the hat to distinguish the quantities associated with T̂L_{s} from those associated with TL_{g}.

The two branches ψ_{q,±}(z, n, t) = ψ_{q}(p, n, t), p = (z, ±), of the Baker-Akhiezer function are linearly independent away from the branch points and their Wronskian is given by

W_{q}(ψ_{q,−}(z), ψ_{q,+}(z)) = \frac{R_{2g+2}^{1/2}(z)}{∏_{j=1}^{g}(z − μ_{j})}.

Here W_{q}(f, g) = a_{q}(n)(f(n)g(n+1)−f(n+1)g(n)) is the usual modified Wronskian.

It is well known that the spectrum of H_{q}(t) is time independent and consists of g + 1 bands

(3.8)  σ(H_{q}) = ⋃_{j=0}^{g}[E_{2j}, E_{2j+1}].

For further information and proofs we refer to [21, Chapter 9].

Finally, let us renormalize the Baker-Akhiezer function

(3.9)  ψ̃_{q}(p, n, t) = \frac{ψ_{q}(p, n, t)}{ψ_{q}(p, 0, t)}

such that ψ̃_{q}(p, 0, t) = 1 and let us define α_{s}(p, t) via

(3.10)  exp (α_{s}(p, t)) = ψ_{q}(p, 0, t) = C(0, 0, t)\frac{θ(\underline{z}(p, 0, t))}{θ(\underline{z}(p, 0, 0))} exp (t ∫_{p_{0}}^{p} Ω_{s}).

4. Commutation methods and N-soliton solutions

In this section we investigate commutation methods when applied to a quasiperiodic finite-gap background solution. In particular, we compute the phase shift (in the Jacobian variety) introduced by the solitons. This can be found for the case of one-dimensional Schrödinger operators in [7] (see also [6] for the elliptic case). The case of Jacobi operators seems to be missing and hence we provide the corresponding results to fill this gap. We want to be rather brief and refer to [8] or [21, Ch. 11] for further details in this connection. Since the time t does not play a role in this section, we will just omit it.

We start by inserting an eigenvalue using the single commutation method. Let H_{q} be a quasi-periodic finite-gap operator and let ψ_{q,±}(z, n) be the branches of the Baker-Akhiezer function which are square summable near ±∞. Fix λ_{1} < inf σ(H_{q}), σ ∈ [−1, 1], define

(4.1)  u_{q,σ}(λ_{1}, n) = \frac{1 + σ}{2} ψ_{q,+}(λ_{1}, n) + \frac{1 − σ}{2} ψ_{q,−}(λ_{1}, n),

and let H_{q,σ} be the (self-adjoint) commuted operator associated with

(4.2)  (τ_{q,σ}f)(n) = a_{q,σ}(n)f(n + 1) + a_{q,σ}(n − 1)f(n − 1) + b_{q,σ}(n)f(n),

where (see [21, Sect. 11.2])

a_{q,σ}(n) = −\frac{\sqrt{a_{q}(n)a_{q}(n + 1)u_{q,σ}(λ_{1}, n)u_{q,σ}(λ_{1}, n + 2)}}{u_{q,σ}(λ_{1}, n + 1)},
b_{q,σ}(n) = λ_{1} − a_{q}(n)(\frac{u_{q,σ}(λ_{1}, n)}{u_{q,σ}(λ_{1}, n + 1)} + \frac{u_{q,σ}(λ_{1}, n + 1)}{u_{q,σ}(λ_{1}, n)})

= b_{q}(n) + ∂^{∗} \frac{a_{q}(n)u_{q,σ}(λ_{1}, n)}{u_{q,σ}(λ_{1}, n + 1)}. (4.3)

H_{q} − λ_{1} and H_{q,σ} − λ_{1} restricted to the orthogonal complements of their corresponding one-dimensional null-spaces are unitarily equivalent and hence

σ_{ac}(H_{q,σ}) = σ_{ac}(H_{q}),  σ_{sc}(H_{q,σ}) = σ_{sc}(H_{q}) = ∅,
σ_{pp}(H_{q,σ}) = \{ \begin{matrix}\{λ_{1}\}, & σ ∈ (−1, 1) \\ ∅, & σ ∈ \{−1, 1\} \end{matrix} .

Lemma 4.1. Let H_{q} be a quasi-periodic finite-gap operator associated with the Dirichlet divisor D\underline{μ̂} and let H_{q,σ}, −1 < σ < 1, be the commuted operator associated with (4.2). Then we have

(4.4)  a_{q,σ}(n) ∼ a_{q,±1}(n), b_{q,σ}(n) ∼ b_{q,±1}(n)  as n → ±∞,

where H_{q,±1} are the quasi-periodic finite-gap operators associated with the Dirichlet divisors D\underline{μ̂}_{±1} defined via

(4.5)  \underline{α}_{p_{0}}(D\underline{μ̂}_{±1}) = \underline{α}_{p_{0}}(D\underline{μ̂}) − \underline{A}_{p_{0}}(p_{1}) − \underline{A}_{p_{0}}(∞_{+}),  p_{1} = (λ_{1}, ±).

Proof. That H_{q,±1} is associated with the divisor D\underline{μ̂}_{±1} is shown in [21, Sect. 11.4] and the asymptotics follow since u_{q,σ}(λ_{1}, n) ∼ \frac{1±σ}{2} u_{q,±1}(λ_{1}, n) as n → ∓∞. □

Similarly, one obtains the following result for the double commutation method. Let λ_{1} ∈ ℝ\σ_{ess}(H_{q}), define (see [21, Sect. 11.6, (2.30)])

c_{q,γ}(λ_{1}, n) = \frac{1}{γ} + ∑_{j=−∞}^{n} ψ_{q,−}(λ_{1}, j)^{2} = \frac{1}{γ} + W_{q,n}(ψ_{q,−}(λ_{1}), ψ̇_{q,−}(λ_{1}))

= \frac{1}{γ} + a_{q}(n)ψ_{q,−}(λ_{1}, n)^{2}ϕ̇_{q,−}(λ_{1}, n), γ ≠ 0, (4.6)

and let H_{q,γ} be the doubly commuted operator associated with

a_{q,γ}(n) = a_{q}(n)\frac{\sqrt{c_{q,γ}(λ_{1}, n − 1)c_{q,γ}(λ_{1}, n + 1)}}{c_{q,γ}(λ_{1}, n)},
(4.7)  b_{q,γ}(n) = b_{q}(n) − ∂^{∗} \frac{a_{q}(n)ψ_{q,−}(λ_{1}, n)ψ_{q,−}(λ_{1}, n + 1)}{c_{q,γ}(λ_{1}, n)}.

Then

Lemma 4.2. Let H_{q} be a quasi-periodic finite-gap operator associated with the Dirichlet divisor D\underline{μ̂} and let H_{q,γ}, 0 < γ < ∞, be the doubly commuted operator associated with (4.7). Then we have

a_{q,γ}(n) = \{ \begin{matrix}a_{q}(n)(1 + O(w(λ_{1})^{2n}))  as & n → −∞ \\ a_{q,∞}(n)(1 + O(w(λ_{1})^{−2n}))  as & n → +∞ \end{matrix} ,
(4.8)  b_{q,γ}(n) = \{ \begin{matrix}b_{q}(n)(1 + O(w(λ_{1})^{2n}))  as & n → −∞ \\ b_{q,∞}(n)(1 + O(w(λ_{1})^{−2n}))  as & n → +∞ \end{matrix} ,

where w(z) = exp(∫_{p_{0}}^{(z,+)} ω_{∞_{+},∞_{−}}) is the quasi-momentum map and H_{q,∞} is the quasi-periodic finite-gap operator associated with the Dirichlet divisor D\underline{μ̂}_{∞} defined via

(4.9)  \underline{α}_{p_{0}}(D\underline{μ̂}_{λ_{1}}) = \underline{α}_{p_{0}}(D\underline{μ̂}) + 2\underline{A}_{p_{0}}(λ̂_{1}),  λ̂_{1} = (λ_{1}, +).

Proof. Since the double commutation method can be obtained via two single commutation steps (see [21, Sect. 11.5]), the result is a consequence of our previous lemma. The asymptotics follow from (4.7) and (3.5). □

Clearly, if we add k eigenvalues λ_{1}, …, λ_{k}, then the asymptotics at +∞ are given by the quasi periodic operator associated with the Dirichlet divisor

(4.10)  \underline{α}_{p_{0}}(D\underline{μ̂}_{λ_{1},...,λ_{k}}) = \underline{α}_{p_{0}}(D\underline{μ̂}) + 2 ∑_{j=1}^{k} \underline{A}_{p_{0}}(λ̂_{j}),  λ̂_{j} = (λ_{j}, +).

In particular, by choosing at least one eigenvalue in each gap, we can attain any prescribed asymptotics in the given isospectral class by Lemma 9.1 in [21].

Remark 4.3. If a_{q}(n, t), b_{q}(n, t) is a quasi-periodic solution of the Toda hierarchy and ψ_{q}(p, n, t) is the corresponding time dependent Baker-Akhiezer function, then a_{q,γ}(n, t), b_{q,γ}(n, t) is a solution of the Toda hierarchy which is centered at

(4.11)  2α(λ_{1})(n − v(λ_{1})t) + ln(γ) = 0,

where

(4.12)  α(λ_{1}) = Re ∫_{p_{0}}^{(λ_{1},−)}ω_{∞_{+},∞_{−}},  v(λ_{1}) = −\frac{1}{α(λ_{1})}Re ∫_{p_{0}}^{(λ_{1},−)}Ω_{s}.

5. Scattering theory

In this section we review scattering theory for Jacobi operators with steplike quasi-periodic finite-gap background in the same isospectral class following [4]. Our only new result in this section will be a full description of the effect of the double commutation method on the scattering data in Lemma 5.5.

More precisely, we will take two quasi-periodic finite-gap operators H_{q}^{±} associated with the sequences a_{q}^{±}, b_{q}^{±} in the same isospectral class,

(5.1)  σ(H_{q}^{+}) = σ(H_{q}^{−}) ≡ Σ = ⋃_{j=0}^{g}[E_{2j}, E_{2j+1}],

but with possibly different Dirichlet data {μ̂_{j}^{±}}_{j=1}^{g}. We will add ± as a superscript to all data introduced in Section 3 to distinguish between the corresponding data of H_{q}^{+} and H_{q}^{−}. To avoid excessive sub/superscripts we abbreviate

(5.2)  ψ_{q}^{±}(z, n) = ψ_{q,±}^{±}(z, n) and ψ̄_{q}^{±}(z, n) = ψ_{q,∓}^{±}(z, n),

that is, ψ_{q}^{±}(z, n) is the solution of H_{q}^{±} decaying near ±∞ and ψ̄_{q}^{±}(z, n) is the solution of H_{q}^{±} decaying near ∓∞. Note that for λ ∈ Σ we have ψ̄_{q}^{±}(λ, n) = \overline{ψ_{q}^{±}(λ, n)}.

Let a(n), b(n) be sequences satisfying

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

and denote the corresponding operator by H.

Theorem 5.1. Assume (5.3). Then there exist solutions ψ_{±}(z, .), z ∈ ℂ, of τψ = zψ satisfying

(5.4)  lim_{n→±∞} |w(z)^{∓n}(ψ_{±}(z, n) − ψ_{q}^{±}(z, n))| = 0,

where w(z) = exp(∫_{p_{0}}^{(z,+)} ω_{∞_{+},∞_{−}}) is the quasi-momentum map.

Theorem 5.2. Assume (5.3). Then we have σ_{ess}(H) = Σ, the point spectrum of H is finite and confined to the spectral gaps of H_{q}^{±}, that is, σ_{p}(H) = {ρ_{j}}_{j=1}^{q} ⊂ ℝ\Σ. Furthermore, the essential spectrum of H is purely absolutely continuous.

Using the fact that ψ_{q}^{±}(p, n) form an orthonormal basis for L^{2}(∂Π_{+}, dω^{±}), where

(5.5)  dω^{±} = \frac{∏_{j=1}^{g}(π − μ_{j}^{±})}{R_{2g+2}^{1/2}}dπ,

we can define

(5.6)  K_{±}(n, m) = 2Re ∫_{Σ} ψ_{±}(λ, n)ψ_{q}^{±}(λ, m)dω^{±}

implying

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

Next we define the coefficients of the scattering matrix via the scattering relations

(5.8)  ψ_{∓}(λ, n) = α_{±}(λ)\overline{ψ_{±}(λ, n)} + β_{±}(λ)ψ_{±}(λ, n),  λ ∈ Σ,

where

(5.9)  α_{±}(λ) = \frac{W (ψ_{±}(λ), ψ_{∓}(λ))}{W (ψ_{±}(λ), \overline{ψ_{±}(λ)})} = \frac{∏_{j=1}^{g}(λ − μ_{j}^{±})}{R_{2g+2}^{1/2}(λ)}W (ψ_{−}(λ), ψ_{+}(λ)),

β_{±}(λ) = \frac{W (ψ_{∓}(λ), \overline{ψ_{±}(λ)})}{W (ψ_{±}(λ), \overline{ψ_{±}(λ)})} = ∓\frac{∏_{j=1}^{g}(z − μ_{j}^{±})}{R_{2g+2}^{1/2}(z)}W (ψ_{∓}(λ), \overline{ψ_{±}(λ)}),

and W_{n}(f, g) = a(n)(f(n)g(n + 1) − f(n + 1)g(n)) denotes the Wronskian. Transmission T_{±}(λ) and reflection R_{±}(λ) coefficients are then defined by

(5.10)  T_{±}(λ) = α_{±}^{−1}(λ),  R_{±}(λ) = \frac{β_{±}(λ)}{α_{±}(λ)} = \frac{W (ψ_{∓}(λ), \overline{ψ_{±}(λ)})}{W (ψ_{±}(λ), ψ_{∓}(λ))}.

The norming constants γ_{±,j} corresponding to ρ_{j} ∈ σ_{p}(H) are given by

(5.11)  γ_{±,j}^{−1} = ∑_{n∈ℤ} |ψ_{±}(ρ_{j}, n)|^{2},  1 ≤ j ≤ q.

Note that γ_{±,j} = 0 if ρ_{j} coincides with a pole μ̂_{ℓ}^{±} ∈ Π_{±} of ψ_{±}(z, .). To avoid this, one could remove the poles by introducing ψ̂_{±}(z, .) as we did in [4]. Since this normalization cancels out in the Gel’fand-Levitan-Marchenko equation and unnecessarily complicates the formulas below, we will allow zero norming constants.

Moreover, ψ_{±}(ρ_{j}, .) = c_{j}^{±}ψ_{∓}(ρ_{j}, .) with c_{j}^{+}c_{j}^{−} = 1.

Lemma 5.3. The coefficients T_{±}(λ), R_{±}(λ) are bounded for λ ∈ Σ, continuous for λ ∈ Σ except at possibly the band edges E_{j}, and fulfill

(5.12)  T_{+}(λ)\overline{T_{−}(λ)} + |R_{±}(λ)|^{2} = 1,  λ ∈ Σ,
(5.13)  T_{±}(λ)\overline{R_{±}(λ)} + \overline{T_{±}(λ)}R_{∓}(λ) = 0,  λ ∈ Σ.

In particular,

(5.14)  |T_{±}(λ)|^{2} ∏_{j=1}^{g} \frac{λ − μ_{j}^{±}}{λ − μ_{j}^{∓}} + |R_{±}(λ)|^{2} = 1,

and hence |R_{±}(λ)|^{2} ≤ 1 with equality only possibly at the band edges {E_{j}}. The transmission coefficients T_{±}(λ) have a meromorphic continuation to ℂ\Σ with simple poles at μ_{j}^{±} if μ̂_{j}^{±} ∈ Π_{∓} and simple poles at ρ_{j},

(5.15)  (Res_{ρ_{j}}T_{±}(λ))^{2} = γ_{+,j}γ_{−,j} \frac{R_{2g+2}(ρ_{j})}{∏_{l=1}^{g}(ρ_{j} − μ_{l}^{±})^{2}} .

In addition, T_{±}(z) ∈ ℝ as z ∈ ℝ\Σ and

(5.16)  T_{±}(∞) = ∏_{n=−∞}^{−1} \frac{a(n)}{a_{q}^{−}(n)} ∏_{n=0}^{∞} \frac{a(n)}{a_{q}^{+}(n)}.

The sets

(5.17)  S_{±}(H) = {R_{±}(λ), λ ∈ Σ; (ρ_{j}, γ_{±,j}), 1 ≤ j ≤ q}

are called left/right scattering data for H.

Theorem 5.4. The kernel K_{±}(n, m) of the transformation operator satisfies the Gel’fand-Levitan-Marchenko equation

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

where

F^{±}(m, n) = 2Re ∫_{Σ} R_{±}(λ)ψ_{q}^{±}(λ, m)ψ_{q}^{±}(λ, n)dω^{±} + ∑_{j=1}^{q} γ_{±,j}ψ_{q}^{±}(ρ_{j}, n)ψ_{q}^{±}(ρ_{j}, m). (5.19)

Note that the apparent poles μ_{ℓ}^{±} cancel with the zeros of dω^{±} and γ_{±,j} at these points.

The operator H can be uniquely reconstructed from S_{±}(H) by solving the Gel’fand-Levitan-Marchenko equation. We refer to [4] for further details.

Finally, we come to our principal new result in this section and investigate the connection with the double commutation method. The scattering data of the operators H, H_{γ} are related as follows.

Lemma 5.5. Let H be a given Jacobi operator satisfying (5.3) and choose ρ_{q+1} ∈ ℝ\σ(H), γ > 0. Then the doubly commuted operator H_{γ}, γ > 0, defined via ψ_{−}(ρ_{q+1}) as in Section 4, satisfies

(5.20) a_{γ}(n) ∼ \{ \begin{matrix}a_{q}^{−}(n) as & n → −∞ \\ a_{q}^{∞}(n) as & n → +∞ \end{matrix} ,  b_{γ}(n) ∼ \{ \begin{matrix}b_{q}^{−}(n) as & n → −∞ \\ b_{q}^{∞}(n) as & n → +∞ \end{matrix} ,

such that (5.3) still holds, where H_{q}^{∞} is associated with Dirichlet data given by

(5.21)  \underline{α}_{p_{0}}(D\underline{μ}^{∞}) = \underline{α}_{p_{0}}(D\underline{μ}^{+}) + 2\underline{A}_{p_{0}}(ρ̂_{q+1}).

It has the scattering data

(5.22)  R_{−,γ}(λ) = R_{−}(λ),
(5.23)  R_{+,γ}(λ) = \frac{θ(\underline{z}^{∞}(λ̂, 0))}{θ(\underline{z}^{+}(λ̂, 0))} \frac{θ(\underline{z}^{+}(λ̂^{∗}, 0))}{θ(\underline{z}^{∞}(λ̂^{∗}, 0))}B(λ, ρ_{q+1})^{2}R_{+}(λ),
(5.24)  T_{+,γ}(z) = C \frac{θ(\underline{z}^{+}(ẑ^{∗}, 0))}{θ(\underline{z}^{∞}(ẑ^{∗}, 0))}B(z, ρ_{q+1})T_{+}(z),
(5.25)  T_{−,γ}(z) = \frac{1}{C} \frac{θ(\underline{z}^{∞}(ẑ, 0))}{θ(\underline{z}^{+}(ẑ, 0))}B(z, ρ_{q+1})T_{−}(z),

where

(5.26)  B(z, ρ) = exp ( − ∫_{E(ρ)}^{(ρ,+)}ωẑ_{}_{ẑ^{∗}})

is the Blaschke factor and λ̂ = (λ, +), ẑ = (z, +). The constant C is given by

(5.27)  C = \sqrt{\frac{θ(\underline{z}^{∞}(∞_{+}, 0))θ(\underline{z}^{∞}(∞_{−}, 0))}{θ(\underline{z}^{+}(∞_{+}, 0))θ(\underline{z}^{+}(∞_{−}, 0))}} > 0.

The norming constants γ_{−,j} corresponding to ρ_{j} ∈ σ_{p}(H), j = 1, …, q, (cf. (5.11)) remain unchanged except for an additional eigenvalue ρ_{q+1} with norming constant γ_{−,q+1} = γ. The norming constants γ_{+,j,γ}, j = 1, …, q, are given by

(5.28)  γ_{+,j,γ} = \frac{1}{C^{2}} (\frac{θ(\underline{z}^{∞}(ρ̂_{j}, 0))}{θ(\underline{z}^{+}(ρ̂_{j}, 0))})^{2} B(ρ_{j}, ρ_{q+1})^{2}γ_{+,j},  ρ̂_{j} = (ρ_{j}, +),

and the additional norming constant γ_{+,q+1,γ} reads (5.29)

γ_{+,q+1,γ} = C^{2} \frac{∏_{j=1}^{g}(ρ_{q+1} − μ_{j}^{+})^{2}}{γR_{2g+2}(ρ_{q+1})} (\frac{θ(\underline{z}^{+}(ρ̂_{q+1}^{∗}, 0))}{θ(\underline{z}^{∞}(ρ̂_{q+1}^{∗}, 0))}T_{+}(ρ_{q+1})Res_{z=ρ_{q+1}}B(z, ρ_{q+1}))^{2}.

Proof. First we show that (5.3) still holds. Note that

a_{γ}(n) = \{ \begin{matrix}a(n)(1 + O(w(ρ_{q+1})^{2n}))  as & n → −∞ \\ a_{∞}(n)(1 + O(w(ρ_{q+1})^{−2n}))  as & n → +∞ \end{matrix} ,
b_{γ}(n) = \{ \begin{matrix}b(n)(1 + O(w(ρ_{q+1})^{2n}))  as & n → −∞ \\ b_{∞}(n)(1 + O(w(ρ_{q+1})^{−2n}))  as & n → +∞ \end{matrix} .

Hence the asymptotics near −∞ are clearly unchanged and for +∞ it suffices to check γ = ∞. By Lemma 5.7 below,

\frac{c_{∞}(ρ_{q+1}, n + 1)}{c_{∞}(ρ_{q+1}, n)} = \frac{c_{∞}(ρ_{q+1}, n + 1)ψ_{+}(ρ_{q+1}, n + 1)}{c_{∞}(ρ_{q+1}, n)ψ_{+}(ρ_{q+1}, n + 1)} = \frac{c_{q,∞}^{+}(ρ_{q+1}, n + 1)}{c_{q,∞}^{+}(ρ_{q+1}, n)}(1 + C(n)),

where ∑_{n∈ℕ} n|C(n)| < ∞. Therefore

(5.30)  a_{∞}(n) = a(n)\frac{\sqrt{c_{∞}(ρ_{q+1}, n − 1)c_{∞}(ρ_{q+1}, n + 1)}}{c_{∞}(ρ_{q+1}, n)} → a_{q,∞}(n)

such that ∑_{n∈ℕ} n|a_{∞}(n) − a_{q,∞}(n)| < ∞ and similarly for b_{∞}(n).

Now we turn to the scattering data. By [21, Lemma 11.19], the Jost solutions ψ_{±,γ}(z, n) of H_{γ} are up to a constant given by (5.31)

u_{±,γ}(z, n) = \frac{c_{γ}(ρ_{q+1}, n)ψ_{±}(z, n) − \frac{1}{z−ρ_{q+1}} ψ_{−}(ρ_{q+1}, n)W_{n−1}(ψ_{−}(ρ_{q+1}), ψ_{±}(z))}{\sqrt{c_{γ}(ρ_{q+1}, n − 1)c_{γ}(ρ_{q+1}, n)}}.

Since this constant is equal to 1 for ψ_{−,γ}(z, n) the fact that R_{−} is unchanged follows from its definition and [21, (11.107)]. The transmission coefficients are reconstructed from R_{−}(λ) using [4, Theorem 3.6] and R_{+,γ}(λ) follows then from

R_{+,γ}(λ) = −\frac{T_{−,γ}(λ)}{\overline{T_{−,γ}(λ)}}R_{−,γ}(λ).

That the norming constants γ_{−,j} are unchanged follows from [21, Lemma 11.14]. For γ_{+,j,γ}, j = 1, …, q + 1, we use (5.15)

γ_{+,j,γ} = \frac{∏_{l=1}^{g}(ρ_{j} − μ_{l}^{±})^{2}(Res_{ρ_{j}}T_{±,γ}(λ))^{2}}{R_{2g+2}(ρ_{j})γ_{−,j}} .

□

Remark 5.6. If we choose ρ = ρ_{j} ∈ σ_{p}(H) and γ = −γ_{−,j}, then the eigenvalue ρ_{j} is removed from the spectrum and it is straightforward to see that an analogous result holds.

The following result used in the previous proof is of independent interest.

Lemma 5.7. Let H be a given Jacobi operator satisfying (5.3). Then for every z ∈ ℂ\Σ and every k ∈ ℕ we have

(5.32) ψ_{∓}(z, n)ψ_{±}(z, n+k)−α_{±}(z)ψ_{q}^{±}(z, n+k)ψ̄_{q}^{±}(z, n) = (k+1)|w(z)|^{k}C_{±}(z, n),

where ∑_{±n∈ℕ} n|C_{±}(z, n)| < ∞.

Similarly, one has

(5.33)  c_{∞}(z, n)ψ_{±}(z, n + k) − α_{±}(z)c_{q,∞}^{±}(z, n)ψ_{q}^{±}(z, n + k) = C̃_{±}(z, n, k),

where ∑_{±n∈ℕ} n|C̃_{±}(z, n, k)| < ∞ and c_{∞}(z, n), c_{q,∞}^{±}(z, n) are defined as in (4.6).

Proof. The proof is an extension of [14, Lemma 3.4] and we will only consider the ’+’ case. First recall that the Green’s functions of H_{q}^{+} and H are given by

G_{q}^{+}(z, n, m) = \frac{ψ̄_{q}^{+}(z, m)ψ_{q}^{+}(z, n)}{W_{q}(ψ̄_{q}^{+}(z), ψ_{q}^{+}(z))}, G(z, n, m) = \frac{ψ_{−}(z, m)ψ_{+}(z, n)}{W (ψ_{−}(z), ψ_{+}(z))}, n ≥ m,

respectively. Considering matrix elements in the second resolvent identity (H − z)^{−1} − (H_{q}^{+} − z)^{−1} = (H − z)^{−1}(H_{q}^{+} − H)(H_{q}^{+} − z)^{−1} we obtain

C_{+}(z, n) = \frac{|w(z)|^{−k}}{k + 1} ∑_{m∈ℤ} W (ψ_{−}(z), ψ_{+}(z))G(z, n, m)(H_{q}^{+} − H)G_{q}^{+}(z, m, n + k)

for z ∈ ℂ\σ(H). Since the poles of W (ψ_{−}(z), ψ_{+}(z))G(z, n, m) at z ∈ σ_{p}(H) are removable, the formula holds for all z ∈ ℂ\Σ. Estimating the right hand side using |G(z, n, m)| ≤ const |w(z)|^{|n−m|} and |G_{q}^{+}(z, n + k, m)| ≤ const |w(z)|^{|n+k−m|} we obtain

|C_{+}(z, n)| ≤ \frac{C|w(z)|^{−k}}{k + 1} ∑_{m∈ℤ} |w(z)|^{|n−m|+|n−m+k|}(2|a(m)−a_{q}^{+}(m)|+|b(m)−b_{q}^{+}(m)|).

We split the sum into three parts

|w(z)|^{|n−m|+|n−m+k|} = \{ \begin{matrix}|w(z)|^{k}|w(z)|^{2|n−m|} & m < n, \\ |w(z)|^{k} & n ≤ m ≤ n + k, \\ |w(z)|^{k}|w(z)|^{2|n−m+k|} & m > n + k. \end{matrix}

Since (5.3) holds for c(m) = 2|a(m)−a_{q}^{+}(m)|+|b(m)−b_{q}^{+}(m)| and we have |w(z)| < 1 for z ∈ ℂ\Σ, we can apply Lemma A.1 to verify ∑_{n∈ℕ} n|C_{+}(z, n)| < ∞.

The claim (5.33) is a consequence of (4.6) and (5.32)

c_{∞}(z, n)ψ_{+}(z, n + k)^{2} = ∑_{j=−∞}^{n} ψ_{−}(z, j)^{2}ψ_{+}(z, n + k)^{2}

= α_{+}(z)^{2}c_{q,∞}^{+}(z, n)ψ_{q}^{+}(z, n + k)^{2} + C̃_{±}(z, n, k)ψ_{q}^{+}(z, n + k),

where C̃_{±}(z, n, k) again can be estimated using Lemma A.1. □

6. Inverse scattering transform

Let a(n, t), b(n, t) be a solution of the Toda hierarchy satisfying

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

Note that by Lemma 2.3 it suffices to check (6.1) for one t_{0} ∈ ℝ (as background take H_{q}^{−}(t) and insert g eigenvalues such that the asymptotics on the other side are given by H_{q}^{+}(t)).

Jost solutions, transmission and reflection coefficients depend now on an additional parameter t ∈ ℝ. The Jost solutions ψ_{±}(z, n, t) are normalized such that

ψ̃_{±}(z, n, t) = ψ̃_{q}^{±}(z, n, t) (1 + o(1)) as n → ±∞,

where (c.f. (3.9))

(6.2)  ψ̃_{q}^{±}(z, n, t) = \frac{ψ_{q}^{±}(z, n, t)}{ψ_{q}^{±}(z, 0, t)} =: exp(−α_{s}^{±}(z, t))ψ_{q}^{±}(z, n, t).

Moreover, we set

(6.3)  exp(ᾱ_{s}^{±}(z, t)) = ψ̄_{q}^{±}(z, 0, t).

Note that we have \overline{exp(α_{s}^{±}(z, t))} = exp(ᾱ_{s}^{±}(z, t)) for λ ∈ Σ.

Transmission and reflection coefficients are then defined via the normalized Jost solutions ψ̃_{±}(z, n, t). Moreover,

(6.4)  σ(H(t)) ≡ σ(H),  t ∈ ℝ.

To avoid the poles of the Baker-Akhiezer function, we will assume that none of the eigenvalues ρ_{j} coincides with a Dirichlet eigenvalue μ_{k}^{±}(0, 0). This can be done without loss of generality by shifting the initial time t_{0} = 0 if necessary.

Remark 6.1. Due to this assumption there is no need to remove these poles for the definition of γ_{±,j}, as we did in [2], [4]. Since the Dirichlet eigenvalues rotate in their gap, the factor needed to remove the poles would only unnecessarily complicate the time evolution of the norming constants. Moreover, these factors would eventually cancel in the Gel’fand-Levitan-Marchenko equation, which is the only interesting object from the inverse spectral point of view in the first place.

Lemma 6.2. Let (a(t), b(t)) be a solution of the Toda hierarchy such that (5.3) holds. The functions

(6.5)  ψ_{±}(z, n, t) = exp(α_{s}^{±}(z, t))ψ̃_{±}(z, n, t)

satisfy

(6.6) H(t)ψ_{±}(z, n, t) = zψ_{±}(z, n, t),  \frac{d}{dt}ψ_{±}(z, n, t) = P̂_{2s+2}(t)ψ_{±}(z, n, t).

Proof. We proceed as in [3], [20, Theorem 3.2]. The Jost solutions ψ_{±}(z, n, t) are continuously differentiable with respect to t by the same arguments as for z (compare [2, Theorem 4.2]) and the derivatives are equal to the derivatives of the Baker-Akhiezer functions as n → ±∞.

For z ∈ ρ(H), the solution u_{±}(z, n, t) of (6.6) with initial condition ψ_{±}(z, n, 0) ∈ ℓ_{±}^{2}(ℤ) remains square summable near ±∞ for all t ∈ ℝ (see [19] or [21, Lemma 12.16]), that is, u_{±}(z, n, t) = C_{±}(t)ψ_{±}(z, n, t). Letting n → ±∞ we see C_{±}(t) = 1. The general result for all z ∈ ℂ now follows from continuity. □

This implies

Theorem 6.3. Let (a(t), b(t)) be a solution of the Toda hierarchy such that (5.3) holds. The time evolution for the scattering data is given by

(6.7)  T_{±}(z, t) = T_{±}(z, 0) exp(α_{s}^{∓}(z, t) − ᾱ_{s}^{±}(z, t)),
(6.8)  R_{±}(λ, t) = R_{±}(λ, 0) exp(α_{s}^{±}(λ, t) − ᾱ_{s}^{±}(λ, t)), λ ∈ Σ,
(6.9)  γ_{±,j}(t) = γ_{±,j}(0) exp(2α_{s}^{±}(ρ_{j}, t)),  1 ≤ j ≤ q.

Proof. Since the Wronskian of two solutions satisfying (6.6) does not depend on n or t (see [19], [21, Lemma 12.15]), we have

T_{±}(z, t) = \frac{W (\overline{ψ̃_{±}(z, t)}, ψ̃_{±}(z, t))}{W (ψ̃_{∓}(z, t), ψ̃_{±}(z, t))} = \frac{exp(α_{s}^{∓}(z, t) + α_{s}^{±}(z, t))}{exp(ᾱ_{s}^{±}(z, t) + α_{s}^{±}(z, t))} \frac{W (\overline{ψ_{±}(z, t)}, ψ_{±}(z, t))}{W (ψ_{∓}(z, t), ψ_{±}(z, t))}
= exp(α_{s}^{∓}(z, t) − ᾱ_{s}^{±}(z, t))T_{±}(z, 0).

The result for R_{±}(λ, t) follows similarly. The time dependence of γ_{±,j}(t) follows from ∥ψ_{±}(ρ_{j}, ., t)∥ = ∥Û_{s}(t, 0)ψ_{±}(ρ_{j}, ., 0)∥ = ∥ψ_{±}(ρ_{j}, ., 0)∥. □

Corollary 6.4. The quantity T_{±}(λ, t)\overline{T_{∓}(λ, t)} = 1 − |R_{±}(λ, t)|^{2}, λ ∈ Σ, does not depend on t.

Another straightforward consequence is:

Theorem 6.5. The time dependence of the kernel of the Gel’fand-Levitan-Marchenko equation is given by

F^{±}(m, n, t) = 2Re ∫_{Σ} R_{±}(λ, 0)ψ_{q}^{±}(λ, m, t)ψ_{q}^{±}(λ, n, t)dω^{±}(0)

+ ∑_{j=1}^{q} γ_{±,j}(0)ψ_{q}^{±}(ρ_{j}, m, t)ψ_{q}^{±}(ρ_{j}, n, t).

Proof. Just employ Theorem 6.3 to rewrite

F^{±}(m, n, t) = ∫_{∂Π_{+}} R_{±}(p, t)ψ̃_{q}^{±}(p, m, t)ψ̃_{q}^{±}(p, n, t)dω^{±}(t)

+ ∑_{j=1}^{q} γ_{±,j}(t)ψ̃_{q}^{±}(ρ_{j}, n, t)ψ̃_{q}^{±}(ρ_{j}, m, t),

where we also use that exp(α_{s}^{±}(λ, t) + ᾱ_{s}^{±}(λ, t)) = ∏_{j=1}^{g} \frac{λ−μ_{j}^{±}(0,t)}{λ−μ_{j}^{±}(0,0)}. □

Finally we note ([19], [21, Section 14.5])

Lemma 6.6. Let (a(t), b(t)) be a solution of the Toda hierarchy such that (5.3) holds. Choose ρ ∈ ℝ\σ(H) and γ > 0. Then (a_{γ}(t), b_{γ}(t)) defined via ψ_{−}(ρ, n, t) using the double commutation method is again a solution of the Toda hierarchy such that (5.3), with H_{q}^{+}(t) accordingly changed, holds.

Acknowledgments. I.E. gratefully acknowledges the extraordinary hospitality of the Faculty of Mathematics of the University of Vienna during two stays in 2006, where parts of this paper were written.

Appendix A. Some estimates

In the proof of Lemma 5.7 we need the following elementary result. Consider a sequence c ∈ ℓ(ℤ) and abbreviate

(A.1)  ∥c∥_{∞} = sup_{n∈ℤ} |c(n)|, ∥c∥_{1} = ∑_{n=0}^{∞} |c(n)|, ∥c∥_{1,1} = ∑_{n=1}^{∞} n|c(n)|.

Lemma A.1. Suppose w is some complex number with |w| < 1 and c ∈ ℓ(ℤ) satisfies ∥c∥_{∞}, ∥c∥_{1,1} < ∞.

Then

∥ ∑_{m=0}^{∞} c(n + m)w^{m}∥_{1,1} ≤ \frac{1}{1 − |w|}∥c∥_{1,1}

and

∥ ∑_{m=0}^{∞} c(n − m)w^{m}∥_{1,1} ≤ \frac{1}{1 − |w|}∥c∥_{1,1} + \frac{|w|}{(1 − |w|)^{2}} ∥c∥_{1} + \frac{|w|^{2}}{(1 − |w|)^{3}} ∥c∥_{∞}.

Proof. The first estimate follows from

∥ ∑_{m=0}^{∞} c(n + m)w^{m}∥_{1,1} = ∑_{n=1}^{∞} n| ∑_{m=0}^{∞} c(n + m)w^{m}|

≤ ∑_{n=0}^{∞} ∑_{m=0}^{∞}(n + m)|c(n + m)||w|^{m}

= ∑_{m=0}^{∞} ∥c∥_{1,1}|w|^{m} = \frac{1}{1 − |w|}∥c∥_{1,1}.

Similarly, the second follows from

∥ ∑_{m=0}^{∞} c(n − m)w^{m}∥_{1,1} = ∑_{n=0}^{∞} n| ∑_{m=0}^{∞} c(n − m)w^{m}|

≤ ∑_{m=0}^{∞} (\frac{m(m − 1)}{2}∥c∥_{∞} + m∥c∥_{1} + ∥c∥_{1,1}) |w|^{m}.

□

References

[1] W. Bulla, F. Gesztesy, H. Holden, and G. Teschl, Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies, Mem. Amer. Math. Soc. 135-641, 1998.

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

[3] 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).

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

[5] M. G. Forest and D. W. McLaughlin, Spectral theory, J. Math. Phs. 23(7), 1248–1277 (1982).

[6] F. Gesztesy, W. Schweiger, and B. Simon, Commutation methods applied to the mKdVequation, Trans. Amer. Math. Soc. 324, No.2, 465–525 (1991).

[7] F. Gesztesy and R. Svirsky, (m)KdV solitons on the background of quasi-periodic finitegap solutions, Mem. Amer. Math. Soc. 118-563, (1995).

[8] F. Gesztesy and G. Teschl, Commutation methods for Jacobi operators, J. Diff. Eqs. 128, 252–299 (1996).

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

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

[11] E. A. Kuznetsov and A. V. Mikhaĭlov, Stability of stationary waves in nonlinear weakly dispersive media, Soviet Phys. JETP 40, No.5, 855–859 (1975).

[12] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure and Appl. Math. 21, 467–490 (1968).

[13] L. Martínez Alonso and E. Olmedilla Moreno, Soliton interaction and prime forms on Riemann surfaces, Phys. Lett. A 188, No.1, 32–38 (1994).

[14] 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.

[15] J. Pełka and J. Zagrodziński, Effective velocity of soliton in the presence of a periodic background, Act. Phys. Poln. A 100, No.6, 871–877 (2001).

[16] J. Rubenstein, Sine-Gordon equation, J. Math. Phys. 11(1), 258–266 (1969).

[17] H. J. Shin, Soliton on a cnoidal wave background in the coupled nonlinear Schrödinger equation, J. Phys. A: Math. Gen. 37, 8017–8030 (2004).

[18] H. J. Shin, The dark soliton on a cnoidal wave background, J. Phys. A: Math. Gen. 38, 3307–3315 (2005).

[19] G. Teschl, On the Toda and Kac-van Moerbeke hierarchies, Math. Z. 231, 325–344 (1999).

[20] G. Teschl, Inverse scattering transform for the Toda hierarchy, Math. Nach. 202, 163– 171 (1999).

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

[22] G. Teschl, Algebro-geometric constraints on solitons with respect to quasi-periodic backgrounds, Bull. London Math. Soc. 39-4, 677–684 (2007).

[23] M. Toda, Theory of Nonlinear Lattices, 2nd enl. ed., Springer, Berlin, 1989.

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

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

Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Wien, Austria, 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, University of Vienna, 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/