PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 6, Pages 1817–1827 S 0002-9939(XX)0000-0

INVERSE SCATTERING TRANSFORM FOR THE TODA HIERARCHY WITH QUASI-PERIODIC BACKGROUND

IRYNA EGOROVA, JOHANNA MICHOR, AND GERALD TESCHL

(Communicated by Joseph A. Ball)

Abstract. We provide a rigorous treatment of the inverse scattering transform for the entire Toda hierarchy in the case of a quasi-periodic finite-gap background solution.

1. Introduction

Since the seminal work of Gardner et al. [7] in 1967 the inverse scattering transform is one of the main tools for solving completely integrable wave equations, and numerous articles have been devoted to this subject since then. However, with only a few exceptions, only the case of short range perturbations of a constant background solution has been investigated. The first to consider a non-constant background seem to be Kuznetsov and A.V. Mikhaĭlov, [8], who informally treated the Korteweg-de Vries equation with the Weierstraß elliptic function as stationary background solution. Formulas for arbitrary periodic background were given by Firsova in [5]. However, the discontinuous behavior at the band edges is not investigated there and neither is the question of when the time dependent scattering data satisfy the hypothesis necessary for the Gel’fand-Levitan-Marchenko theory (see, e.g., [10] for the constant background case). Moreover, the normalization chosen there blows up when the eigenvalues lie at certain positions in the gaps.

Our aim in the present paper is to provide a rigorous treatment of the inverse scattering transform for the Toda hierarchy in the case of quasi-periodic finitegap background solutions (including the case of arbitrary periodic solutions). We include a discussion of the problems arising from the poles of the Baker-Akhiezer functions at the Dirichlet eigenvalues (which are absent in the constant background case). In the constant background case this was first done for the Toda equation by Flaschka in [6] on an informal level. The first rigorous treatment for the entire Toda hierarchy was given by one of us in [12] (see also [4], [15], [14]). For further results on solutions of the Toda equation with non-trivial spatial asymptotics see [1].

After introducing the Toda hierarchy in Section 2, we will first show that a solution will stay close to a given background solution in Section 3. This result

\begin{matrix}c \\ ◯ \end{matrix}2007 American Mathematical Society

Received by the editors December 1, 2005 and, in revised form, February 7, 2006.

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

Key words and phrases. Inverse scattering, Toda hierarchy, periodic.

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

implies that a short-range perturbation of a quasi-periodic finite-gap solution will stay short range for all time, and it shows that the time-dependent scattering data satisfy the hypothesis necessary for the Gel’fand-Levitan-Marchenko theory. This result constitutes the main technical ingredient for the inverse scattering transform. In Section 4 we review some necessary facts on quasi-periodic finite-gap solutions and in Section 5 we compute the time dependence of the scattering data and discuss its dynamics. Finally we establish the conserved quantities of the Toda hierarchy in this context.

2. The Toda hierarchy

In this section we introduce the Toda hierarchy using the standard Lax formalism ([9]). We first review some basic facts from [2] (see also [14]).

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.

Finally, 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).

3. The initial value problem

First of all we recall the basic existence and uniqueness theorem for the Toda hierarchy (see, e.g., [12], [13], or [14, Section 12.2]).

Theorem 3.1. 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}).

In [12] it was shown that solutions which are asymptotically close to the constant solution at the initial time stay close for all time. Our first aim is to extend this result to include perturbations of quasi-periodic finite-gap solutions. In fact, we will even be a bit more general.

Lemma 3.2. Suppose a(n, t), b(n, t) and ā(n, t), b̄(n, t) are two arbitrary bounded solutions of the Toda hierarchy satisfying (3.1) for one t_{0} ∈ ℝ. Then (3.1) holds for all t ∈ ℝ, that is,

(3.1)  ∑_{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)}|) < ∞.

Proof. Without loss of generality we assume that t_{0} = 0. Consider the expression

∥(a(t), b(t))∥_{∗} = ∑_{n∈ℤ} w(n)(|a(n, t) − ā(n, t)| + |b(n, t) − b̄(n, t)|)

which remains finite at least for small t since there is a local solution of the Toda hierarchy with respect to ∥.∥_{∗}. We claim that the following estimate holds:

∑_{n∈ℤ} w(n)|g_{r}(n, t) − ḡ_{r}(n, t)| ≤ C_{r}∥(a(t), b(t))∥_{∗},
(3.2)  ∑_{n∈ℤ} w(n)|h_{r}(n, t) − h̄_{r}(n, t)| ≤ C_{r}∥(a(t), b(t))∥_{∗},

where C_{r} = C(∥H(0)∥, ∥H̄(0)∥) is a positive constant due to Hypothesis H.2.1. Let us prove (3.2) by induction on r. It suffices to consider the case where c_{j} = 0, 1 ≤ j ≤ r, since all involved sums are finite. In this case [14, Lemma 6.4] shows that g_{j}(n, t), h_{j}(n, t) can be recursively computed from g_{0}(n, t) = 1, h_{0}(n, t) = 0 via

g_{j+1}(n, t) = \frac{1}{2}(h_{j}(n, t) + h_{j}(n − 1, t)) + b(n, t)g_{j}(n, t),
h_{j+1}(n, t) = 2a(n, t)^{2} ∑_{l=0}^{j} g_{j−l}(n, t)g_{l}(n + 1, t) − \frac{1}{2} ∑_{l=0}^{j} h_{j−l}(n, t)h_{l}(n, t).

Hence

g_{1}(n, t) − ḡ_{1}(n, t) = b(n, t) − b̄(n, t), h_{1}(n, t) − h̄_{1}(n, t) = 2(a(n, t)^{2} − ā(n, t)^{2})

and

g_{r+1}(n, t) − ḡ_{r+1}(n, t) = (b(n, t) − b̄(n, t))g_{r}(n, t) + b̄(n, t)(g_{r}(n, t) − ḡ_{r}(n, t))

+ \frac{1}{2}(h_{r}(n, t) − h̄_{r}(n, t) + h_{r}(n − 1, t) − h̄_{r}(n − 1, t)),

h_{r+1}(n, t) − h̄_{r+1}(n, t) = \frac{1}{2} ∑_{l=0}^{r} (h_{r−l}(n, t)h_{l}(n, t) − h̄_{r−l}(n, t)h̄_{l}(n, t))

+ 2a(n, t)^{2} ∑_{l=0}^{r} g_{r−l}(n, t)g_{l}(n + 1, t) − 2ā(n, t)^{2} ∑_{l=0}^{r} ḡ_{r−l}(n, t)ḡ_{l}(n + 1, t).

Since both sequences a, b and ā, b̄ are solutions of the Toda hierarchy, (2.8) yields

ȧ(t) − ˙ ā(t) = a(t)(g_{r+1}^{+}(t) − g_{r+1}(t)) − ā(t)(ḡ_{r+1}^{+}(t) − g_{r+1}(t))

= (a(t) − ā(t))(ḡ_{r+1}^{+}(t) − ḡ_{r+1}(t)) − a(t)(ḡ_{r+1}^{+}(t) − g_{r+1}^{+}(t) + g_{r+1}(t) − ḡ_{r+1}(t))

and thus

|a(n, t) − ā(n, t)| ≤ |a(n, 0) − ā(n, 0)| + C_{1} ∫_{0}^{t} |a(n, s) − ā(n, s)|ds

+ ∥H(0)∥ ∫_{0}^{t} (|g_{r+1}(n + 1, s) − ḡ_{r+1}(n + 1, s)| + |g_{r+1}(n, s) − ḡ_{r+1}(n, s)|)ds,

|b(n, t) − b̄(n, t)| ≤ |b(n, 0) − b̄(n, 0)|

+ ∫_{0}^{t}(|h_{r+1}(n, s) − h̄_{r+1}(n, s)| + |h_{r+1}(n − 1, s) − h̄_{r+1}(n − 1, s)|)ds.

By (3.2), ∥(a(t), b(t))∥_{∗} ≤ ∥(a(0), b(0))∥_{∗} + C ∫_{0}^{t} ∥(a(s), b(s))∥_{∗}ds, where C = (C_{1} + 2∥H(0)∥ + 2)C_{r+1}. Finally, applying Gronwall’s inequality

∥(a(t), b(t))∥_{∗} ≤ ∥(a(0), b(0))∥_{∗} exp (Ct)

finishes the proof. □

4. 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 [2] or [14]).

Let 𝕄 be the Riemann surface associated with the following function:

(4.1)  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.

Now pick g numbers (the Dirichlet eigenvalues)

(4.2)  (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},
(4.3)  \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 the 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 [14, 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}},

(4.4)  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 [14, 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_{∞_{+},∞_{−}}),
(4.5)  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,

(4.6)  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 [14, 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)
(4.7)  = P̂_{q,2s+2}(t)ψ_{q}(p, n, t)(n),

where we use the hat to distinguish the quantities associated with T̂L_{s} from those associated with TL_{g}.

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

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

For further information and proofs we refer to [14, Chapter 9].

Next let us renormalize the Baker-Akhiezer function

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

(4.9)  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}).

One can show ([14], (13.49))

exp (α_{s}(p, t)) = exp ( ∫_{0}^{t} (2a_{q}(0, x)Ĝ_{s}(p, 0, x)ϕ_{q}(p, 0, x) − Ĥ_{s+1}(p, 0, x))dx)

= \sqrt{\frac{G_{g}(p, 0, t)}{G_{g}(p, 0, 0)}} exp (R_{2g+2}^{1/2}(p) ∫_{0}^{t} \frac{Ĝ_{s}(p, 0, x)}{G_{g}(p, 0, x)}dx), (4.10)

where

(4.11)  G_{g}(z, n, t) = ∏_{j=1}^{g}(z − μ_{j}(n, t)).

However, let us emphasize that the integrals in (4.10) are only well defined until we hit the first pole of the integrand. After that time they have to be interpreted as principal values (compare also Remark 5.4(i)), respectively, (4.9) has to be used.

In addition, observe

exp (α_{s,+}(z, t) + α_{s,−}(z, t)) = \frac{G_{g}(z, 0, t)}{G_{g}(z, 0, 0)},
(4.12)  exp (α_{s,+}(z, t) − α_{s,−}(z, t)) = exp (2R_{2g+2}^{1/2}(z) ∫_{0}^{t} \frac{Ĝ_{s}(z, 0, x)}{G_{g}(z, 0, x)}dx),

where α_{s,±} are the two branches of α_{s}. For λ ∈ σ(H_{q}) we have

(4.13)  \overline{α_{s,±}(λ, t)} = α_{s,∓}(λ, t),

but note that α_{s,∓}(λ, t) ≠ ∓α_{s,±}(λ, t) unless g = 0.

5. Inverse scattering transform

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

(5.1)  ∑_{n∈ℤ} |n|(|a(n, t) − a_{q}(n, t)| + |b(n, t) − b_{q}(n, t)|) < ∞

for one (and hence for any) t_{0} ∈ ℝ. In [3] (see also [16]) we develop scattering theory for the Jacobi operator H associated with a(n), b(n). Jost solutions, transmission and reflection coefficients now depend on an additional parameter t ∈ ℝ. More precisely,

(5.2) σ(H(t)) ≡ σ(H), σ_{ess}(H) = σ(H_{q}), σ_{p}(H) = {ρ_{j}}_{j=1}^{q} ⊆ ℝ\σ(H_{q}),

where q ∈ ℕ is finite. 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 → ±∞.

Transmission T (λ, t) and reflection R_{±}(λ, t) coefficients are defined via the scattering relations

(5.3)  T (λ, t)ψ̃_{±}(λ, n, t) = \overline{ψ̃_{∓}(λ, n, t)} + R_{∓}(λ, t)ψ̃_{∓}(λ, n, t),  λ ∈ σ(H_{q}),

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

(5.4)  γ_{±,j}(t)^{−1} = ∑_{n∈ℤ} |ψ̃_{±}(ρ_{j}, n, t)|^{2}.

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 5.1. Due to this assumption there is no need to remove these poles for the definition of γ_{±,j}, as we did in [3]. 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. See Remark 5.4 below.

Lemma 5.2. Let (a(t), b(t)) be a solution of the Toda hierarchy such that (5.1) holds. The functions

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

satisfy

(5.6) 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 [12, 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 [3, 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 (5.6) with initial condition ψ_{±}(z, n, 0) ∈ ℓ_{±}^{2}(ℤ) remains square summable near ±∞ for all t ∈ ℝ (see [13] or [14, 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 5.3. Let (a(t), b(t)) be a solution of the Toda hierarchy such that (5.1) holds. The time evolution for the scattering data is given by

T (λ, t) = T (λ, 0),
R_{±}(λ, t) = R_{±}(λ, 0) exp(±(α_{s,+}(λ, t) − α_{s,−}(λ, t))),
(5.7)  γ_{±,j}(t) = γ_{±,j}(0) exp(2α_{s,±}(ρ_{j}, t)),  1 ≤ j ≤ q.

Proof. Since \overline{α_{s,±}(λ, t)} = α_{s,∓}(λ, t) by (4.13), we have

T (λ, t) = \frac{W (ψ̃_{−}(λ, t), \overline{ψ̃_{−}(λ, t)})}{W (ψ̃_{−}(λ, t), ψ̃_{+}(λ, t))}

= \frac{exp(α_{s,−}(λ, t)) exp(α_{s,+}(λ, t))}{exp(α_{s,−}(λ, t))\overline{exp(α_{s,−}(λ, t))}} \frac{W (ψ_{−}(λ, t), \overline{ψ_{−}(λ, t)})}{W (ψ_{−}(λ, t), ψ_{+}(λ, t))} = T (λ, 0).

Here we used that the Wronskian W_{n}(f, g) = a(n)(f(n)g(n + 1) − f(n + 1)g(n)) of two solutions satisfying (5.6) does not depend on n or t (see [13], [14, Lemma 12.15]). Similarly,

R_{±}(λ, t) = \frac{exp(α_{s,±}(λ, t)) exp(α_{s,∓}(λ, t))}{exp(α_{s,∓}(λ, t)) exp(\overline{α_{s,±}(λ, t)})} \frac{W (ψ_{∓}(λ, t), \overline{ψ_{±}(λ, t)})}{W (ψ_{±}(λ, t), ψ_{∓}(λ, t))}
= exp(α_{s,±}(λ, t) − α_{s,∓}(λ, t))R_{±}(λ, 0).

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)∥. □

A few remarks concerning this theorem are in order.

Remark 5.4. (i) First of all note that the time dependence of R_{±}(λ, t) is not continuous at the band edges (except the lowest and highest). In fact, the integrand in (4.12) has a pole whenever μ_{j}(0, t) hits a band edge. Suppose that μ_{j}(0, t) gets close to E_{2j} at t = t_{1}. Then (assuming Ĝ_{s}(E_{2j}, 0, t_{1}) ≠ 0 for simplicity)

μ_{j}(0, t) = E_{2j} − |ν_{j}|^{2} (t − t_{1})^{2} + O(t − t_{1})^{3}, ν_{j} = \frac{Ĝ_{s}(E_{2j}, 0, t_{1}) ∏_{k≠2j} \sqrt{E_{2j} − E_{k}}}{∏_{l≠j}(μ_{j}(0, t_{1}) − μ_{l}(0, t_{1}))} .

As long as t < t_{1} we have

lim_{λ↓E_{2j}}(α_{s,+}(λ, t) − α_{s,−}(λ, t)) = 0.

But at t = t_{1} we obtain (setting λ = E_{2j} + ε^{2} and using ν_{j} = (−1)^{g−j}i|ν_{j}|)

lim_{λ↓E_{2j}}(α_{s,+}(λ, t) − α_{s,−}(λ, t)) = lim_{λ↓E_{2j}} 2R_{2g+2}^{1/2}(λ) ∫_{0}^{t_{1}} \frac{Ĝ_{s}(λ, 0, x)}{G_{g}(λ, 0, x)}dx

= lim_{ε↓0} 2 ∫_{0}^{t_{1}} \frac{ν_{j}ε}{ε^{2} + |ν_{j}|^{2} (x − t_{1})^{2}} dx = 2(−1)^{g−j}i lim_{ε↓0} arctan(\frac{t_{1}|ν_{j}|}{ε}) = (−1)^{g−j}iπ.

This is in agreement with the conditions

(5.8)  \begin{matrix}lim_{λ→E_{l}} R_{2r+2}^{1/2}(λ)\frac{R_{±}(λ,t)+1}{T (λ)} = 0, & E_{l} ≠ μ_{j}(0, t), \\ lim_{λ→E_{l}} R_{2r+2}^{1/2}(λ)\frac{R_{±}(λ,t)−1}{T (λ)} = 0, & E_{l} = μ_{j}(0, t), \end{matrix}

found in [3].

(ii) The norming constants γ_{±,j}(t) will vanish whenever (ρ_{j}, ±) = μ̂_{k}(0, t). This is important, since the norming constants must cancel the pole of ψ̃_{q,±}(ρ_{j}, n, t) at these points such that the Gel’fand-Levitan-Marchenko equation remains well defined.

In fact, these poles come from the normalization ψ̃_{q,±}(z, 0, t) = 1 and this normalization is natural in the constant background case since it renders ψ̃_{q,±}(z, n, t) time-independent. In the (quasi-)periodic background case this is no longer true, and it seems more natural to allow a time-dependent normalization and replace ψ̃_{q,±}(z, n, t) by the time-dependent Baker-Akhiezer functions ψ_{q,±}(z, n, t) rendering the norming constants and the reflection coefficient time-independent.

(iii) Finally let us discuss the connection with the different normalizations for γ_{±,j} used in [3], respectively [5].

In [3] we have multiplied ψ_{q,±}(λ, n, t) by a factor to cancel the poles at the Dirichlet eigenvalues. However, since the Dirichlet eigenvalues change sheets during time evolution, this normalization renders ψ̂_{q,±}(λ, n, t) discontinuous with respect to t and complicates the time evolution of the corresponding norming constants:

(5.9)  γ̂_{±,j}(t) = γ̂_{±,j}(0) exp ( ± 2R_{2g+2}^{1/2}(ρ_{j}) ∫_{0}^{t} \frac{Ĝ_{s}(ρ_{j}, 0, x)}{G_{g}(ρ_{j}, 0, x)}dx

± ∑_{l=1}^{g} ∫_{0}^{t} \frac{2Ĝ_{s}(μ_{l}(x), 0, x)R_{2g+2}^{1/2}(μ_{l}(x))}{(ρ_{j} − μ_{l}(x)) ∏_{k≠l}(μ_{l}(x) − μ_{k}(x))}dx).

This clearly justifies our decision to use the freedom of choosing the initial time to avoid the poles in the first place.

If H_{q} is periodic with period N, the normalization chosen in [5] is to divide ψ_{q,±}(λ, n, t) by the square root of \frac{1}{N} ∑_{n=1}^{N} ψ_{q,+}(z, n)ψ_{q,−}(z, n). To get a welldefined real-valued result this object should be positive. However, since one can compute (see [3])

(5.10)  ∑_{n=1}^{N} ψ_{q,+}(z, n)ψ_{q,−}(z, n) = N ∏_{j=1}^{N−1} \frac{z − λ_{j}}{z − μ_{j}} ,

where λ_{j} ∈ [E_{2j−1}, E_{2j}] are the zeros of ω_{∞_{−},∞_{+}}, this factor can be negative or even zero. While this choice does give a simple time dependence

(5.11)  γ̃_{±,j}(t) = γ̃_{±,j}(0) exp(α_{s,±}(ρ_{j}, t) − α_{s,∓}(ρ_{j}, t)),

it does not cancel the poles at the Dirichlet eigenvalues (it just renders poles into square root singularities), plus it introduces additional singularities at λ_{j}. In particular, it is unsuitable if one of the eigenvalues coincides with some λ_{j}.

In summary, since Lemma 3.2 ensures that (5.1) remains valid for all t once it holds for the initial condition, we can compute R_{±}(λ, 0) and γ_{±,j}(0) from (a(n, 0), b(n, 0)) and then solve the Gel’fand-Levitan-Marchenko (GLM) equation to obtain the sequences (a(n, t), b(n, t)) as in [3]. In this respect the most important ingredient is the time dependence of the kernel of the GLM equation which follows from Theorem 5.3.

Theorem 5.5. The time dependence of the kernel of the Gel’fand-Levitan-Marchenko equation is given by

F^{±}(m, n, t) = \frac{1}{π} Re ∫_{λ∈σ(H_{q})} R_{±}(λ, 0)ψ_{q,±}(λ, m, t)ψ_{q,±}(λ, n, t)\frac{G_{g}(λ, 0, 0)}{R_{2g+2}^{1/2}(λ)}dλ

+ ∑_{j=1}^{q} γ_{±,j}(0)ψ_{q,±}(ρ_{j}, m, t)ψ_{q,±}(ρ_{j}, n, t). (5.12)

Finally note that since the transmission coefficient is time independent we obtain

Corollary 5.6. Let (a(t), b(t)) be a solution of the Toda hierarchy such that (5.1) holds. Then the quantities

(5.13)  A = ∏_{j=−∞}^{∞} \frac{a(j, t)}{a_{q}(j, t)}

and τ_{j} = tr(H^{j}(t) − H_{q}(t)^{j}), that is,

τ_{1} = ∑_{n∈ℤ} b(n, t) − b_{q}(n, t)

τ_{2} = ∑_{n∈ℤ} 2(a(n, t)^{2} − a_{q}(n, t)^{2}) + (b(n, t)^{2} − b_{q}(n, t)^{2}),

etc., are conserved quantities.

Proof. This follows from the fact that the transmission coefficient is the perturbation determinant (in the sense of Krein) of the pair H_{q} and H (see [11]). □

Acknowledgments

We thank the referee for several valuable suggestions for improving the presentation.

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

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

Faculty of Mathematics, 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, 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/