INVERSE SCATTERING TRANSFORM FOR THE TODA HIERARCHY WITH STEPLIKE FINITE-GAP BACKGROUNDS

IRYNA EGOROVA, JOHANNA MICHOR, AND GERALD TESCHL

Abstract. We provide a rigorous treatment of the inverse scattering transform for the entire Toda hierarchy for solutions which are asymptotically close to (in general) different finite-gap solutions as n → ±∞.

1. Introduction

The Toda lattice is one of the most prominent discrete integrable wave equations. In particular, it can be solved via the inverse scattering method. For the classical case, where the solution is asymptotically equal to the (same) constant solution, this is of course well understood and covered in several monographs (e.g.) [12], [36], or [33]. The corresponding long-time asymptotics were first computed by Novokshenov and Habibullin [29] and were later made rigorous by Kamvissis [16] under the additional assumption that no solitons are present (the case of solitons was recently added in [23]; see also the review [22]).

The inverse scattering transform for the entire Toda hierarchy in the case of a finite-gap background was solved only recently by us in [8] as a continuation of [31]. Similar results were obtained by Khanmamedov [21]. Long-time asymptotics for such solutions have been given by Kamvissis and Teschl for the case without solitons [18], [19], [20] and by Krüger and Teschl for the case with solitons [24] (for related trace formulas and conserved quantities see [27]).

In this respect it is important to mention that even the important case of a one-soliton solution on a finite-gap background has different spatial asymptotics as n → ±∞ and hence is not covered by the above results (see [11], [34]). Hence this clearly raises the need to extend the results from [8] to the case of solutions which are asymptotically equal to (in general) different finite-gap solutions as n → ±∞.

In fact, the simplest case, where the solution is asymptotically equal to two different constant solutions, has already attracted considerable interest in the past. The first to solve the corresponding Cauchy problem (in the case of rapid decay with respect to the background) seems to be Oba [30]. Moreover, the long-time asymptotics were considered in [1], [3], [4], [14], [15], [17], [37].

Our aim here is to fill this gap and to provide a treatment of the inverse scattering transform for the entire Toda hierarchy in the case of steplike quasi-periodic finitegap backgrounds. Note that since we treat the entire Toda hierarchy, our results also cover the Kac–van Moerbeke hierarchy as a special case [28].

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

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

Research supported by the Austrian Science Fund (FWF) under Grants No. Y330, V120.

J. Math. Phys. 50, 103522 (2009).

Finally, we remark that the corresponding result for the Korteweg–de Vries equation is much more involved and was only recently solved by Egorova, Grunert, and Teschl [6] under some additional restrictions on the spectra of the background operators.

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 implies that a short-range perturbation of a steplike 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 [26]. 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.

2. The Toda hierarchy

In this section we introduce the Toda hierarchy using the standard Lax formalism ([25]). We first review some basic facts from [2] (see also [13], [33]).

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},

where ⟨δ_{m}, Aδ_{n}⟩ denote the matrix elements of an operator A with respect to the standard basis, 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(n, t) − a(n, t)(g_{r+1}(n + 1, t) − g_{r+1}(n, t)) = 0,
(2.8)  TL_{r}(a(t), b(t))_{2} = ḃ(n, t) − (h_{r+1}(n, t) − h_{r+1}(n − 1, t)) = 0,

where the dot denotes a derivative with respect to t. 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., [31], [32], or [33, 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 [31] 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. Set

(3.1)  ∥(a, b)∥_{w,p} = \begin{cases}( ∑_{n∈ℤ} w(n)(|a(n)|^{p} + |b(n)|^{p}))^{1/p} , & 1 ≤ p < ∞, \\ sup_{n∈ℤ} w(n)(|a(n)| + |b(n)|), & p = ∞. \end{cases}

Then

Lemma 3.2. Let w(n) ≥ 1 be some weight with sup_{n}(|\frac{w(n+1)}{w(n)}|+|\frac{w(n)}{w(n+1)}|) < ∞ and fix some 1 ≤ p ≤ ∞. Suppose a(n, t), b(n, t) and a_{±}(n, t), b_{±}(n, t) are arbitrary bounded solutions of the Toda hierarchy and abbreviate

(3.2)  ā(n, t) = \begin{cases}a_{+}(n, t), & n ≥ 0, \\ a_{−}(n, t), & n < 0, \end{cases}  b̄(n, t) = \begin{cases}b_{+}(n, t), & n ≥ 0, \\ b_{−}(n, t), & n < 0. \end{cases}

Then, if

(3.3)  ∥(a(t) − ā(t), b(t) − b̄(t))∥_{w,p} < ∞

holds for one t = t_{0} ∈ ℝ, then it holds for all t ∈ ℝ.

Proof. Without loss of generality we assume that t_{0} = 0. Let us consider the differential equation for the differences δ(n, t) = (a(n, t) − ā(n, t), b(n, t) − b̄(n, t)) in the Banach space of pairs of bounded sequences δ = (δ_{1}, δ_{2}) for which the norm ∥δ∥_{w,p} is finite. We claim that δ satisfies an inhomogeneous linear differential equation of the form

δ̇(t) = ∑_{|j|≤r+1} A_{r,j}(t)(S^{+})^{j}δ(t) + B_{r}(t)

(see e.g. [5] for the theory of ordinary differential equations in Banach spaces). Here S^{±}(δ_{1}(n, t), δ_{2}(n, t)) = (δ_{1}(n ± 1, t), δ_{2}(n ± 1, t)) are the shift operators,

A_{r,j}(n, t) = (\begin{matrix}A_{r,j}^{11}(n, t) & A_{r,j}^{12}(n, t) \\ A_{r,j}^{21}(n, t) & A_{r,j}^{22}(n, t) \end{matrix}) ,

are multiplication operators with bounded two by two matrix-valued sequences, and

B_{r}(n, t) = (\begin{matrix}B_{r,1}(n, t) \\ B_{r,2}(n, t) \end{matrix})

is a vector in our Banach space with B_{r,i}(n, t) = 0 for |n| > ⌊\frac{r}{2}⌋ + 1. All entries of A_{r,j}(t) and B_{r}(t) are polynomials with respect to (a(n + j, t), b(n + j, t)), (a_{±}(n + j, t), b_{±}(n + j, t)), |j| ≤ ⌊\frac{r}{2}⌋ + 1. Moreover, by our assumption the shift operators are continuous,

∥S^{±}∥ = \begin{cases}sup_{n∈ℤ} |\frac{w(n)}{w(n±1)}|^{1/p}, & p ∈ [1, ∞), \\ sup_{n∈ℤ} |\frac{w(n)}{w(n±1)}|, & p = ∞, \end{cases}

and same is true for the multiplication operators A_{r,j}(t) whose norms depend only on the supremum of the entries by Hölder’s inequality, that is, on the sup norms of (a(t), b(t)) and (a_{±}(t), b_{±}(t)). Finally, recall that by unitary equivalence of the operator family H(t), respectively H_{±}(t), we have a uniform bound of the sup norm sup_{n}(|a(n, t)|+|b(n, t)|) ≤ 2∥H(t)∥ = 2∥H(0)∥, respectively sup_{n}(|a_{±}(n, t)|+ |b_{±}(n, t)|) ≤ 2∥H_{±}(t)∥ = 2∥H_{±}(0)∥. Consequently, there is a constant such that ∑_{|j|≤r+1} ∥A_{r,j}(t)∥∥(S^{+})^{j}∥ ≤ C_{r}. Moreover, we will show below that the vector B_{r}(t) has only finitely many nonzero entries and thus ∥B_{r}(t)∥_{w,p} ≤ D_{r}, where the constant again depends only on the sup norms of (a(t), b(t)) and (a_{±}(t), b_{±}(t)). Hence

∥δ(t)∥_{w,p} ≤ ∥δ(0)∥_{w,p} + ∫_{0}^{t} (C_{r}∥δ(s)∥_{w,p} + D_{r})

and Gronwall’s inequality implies

∥δ(t)∥_{w,p} ≤ ∥δ(0)∥_{w,p}e^{C_{r}t} + \frac{D_{r}}{C_{r}} (e^{C_{r}t} − 1).

It remains to show existence of the above differential equation. This will follow once we show that g_{r+1}(t) − ḡ_{r+1}(t) and h_{r+1}(t) − h̄_{r+1}(t) can be written as a linear combination of shifts of δ with the coefficients depending only on (a(t), b(t)) and (a_{±}(t), b_{±}(t)). The fact that (ā, b̄) does not solve TL_{r} only affects finitely many terms and gives rise to the inhomogeneous term B_{r}(t) which is nonzero only for a finite number of terms.

To see that g_{r+1}(t) − ḡ_{r+1}(t) and h_{r+1}(t) − h̄_{r+1}(t) can be written as a linear combination of shifts of δ we can use induction on r. It suffices to consider the homogenous case where c_{j} = 0, 1 ≤ j ≤ r, since all involved sums are finite. In this case [33, 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)

and similarly for ḡ_{j}(n, t), h̄_{j}(n, t). Hence the claim follows.

Finally, observe that since w(n) ≥ 1 this solution is bounded and hence coincides with the solution of the Toda equation from Theorem 3.1. □

For closely related results we also refer to [35].

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], [13], or [33]).

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

(4.1)  H_{q}^{±}(t)f(n) = a_{q}^{±}(n, t)f(n + 1) + a_{q}^{±}(n − 1, t)f(n − 1) + b_{q}^{±}(n, t)f(n)

in ℓ^{2}(ℤ) associated with the Riemann surface of the square root

(4.2)  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}^{±}(t) are uniquely determined by fixing a Dirichlet divisor ∑_{j=1}^{g^{±}}(μ_{j}^{±}(t), σ_{j}^{±}(t)), where μ_{j}^{±}(t) ∈ [E_{2j−1}^{±}, E_{2j}^{±}] and σ_{j}^{±}(t) ∈ {−1, 1}. The time evolution of the Dirichlet divisor is determined by the Dubrovin equations (cf. [33, (13.2)])and linearized by the Abel map (cf. [33, Sect. 13.2]). The spectra of H_{q}^{±}(t) consist of g_{±} + 1 bands

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

We will identify the set ℂ \ σ(H_{q}^{±}(t)) with the upper sheet of the Riemann surface. Associated with H_{q}^{±}(t) are the Weyl solutions

(4.4)  ψ_{q}^{±}(z, n, t) ∈ ℓ^{2}(±ℕ)

normalized such that ψ_{q}^{±}(z, 0, t) = 1. We will use the convention that for λ ∈ σ_{±} we set ψ_{q}^{±}(λ, n, t) = lim_{ϵ↓0} ψ_{q}^{±}(λ + iϵ). Then

(4.5)  ψ̂_{q}^{±}(z, n, t) = exp (α_{r}^{±}(z, t))ψ_{q}^{±}(z, n, t)

satisfies

(4.6)  H_{q}^{±}(t)ψ̂_{q}^{±}(z, n, t) = zψ̂_{q}^{±}(z, n, t),
(4.7)  \frac{d}{dt}ψ̂_{q}^{±}(z, n, t) = P_{q,2r+2}^{±}(t)ψ̂_{q}^{±}(z, n, t),

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

where ([33], (13.47))

(4.8)  α_{r}^{±}(z, t) = ∫_{0}^{t} (2a_{q}^{±}(0, s)G_{q,r}^{±}(z, 0, s)ψ_{q}^{±}(z, 1, s) − H_{q,r+1}^{±}(z, 0, s))ds.

Note that the integrand in this last expression might have poles if z lies in one of the spectral gaps [E_{2j−1}^{±}, E_{2j}^{±}]. Hence one has to understand α_{r}^{±}(z, t) as a limit from z ∈ ℂ\ℝ for such values of z. Alternatively one can use the expression in terms of Riemann theta functions. Moreover, exp (α_{r}^{±}(z, t)) has simple poles at μ_{j}^{±}(0) and simple zeros at μ_{j}^{±}(t). We refer to the discussion in [8] for further details.

5. Inverse scattering transform

Fix two quasi-periodic finite-gap solutions a_{q}^{±}(n, t), b_{q}^{±}(n, t) as in the previous section. Let a(n, t), b(n, t) be a solution of the Toda hierarchy satisfying

(5.1)  ∑_{n=0}^{±∞}(1 + |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 [10] (see also [7], [9], [38]) we have developed scattering theory for the Jacobi operator H(t) associated with a(n, t), b(n, t). Jost solutions, transmission and reflection coefficients now depend on an additional parameter t ∈ ℝ. The essential spectrum of H(t) is (absolutely) continuous and

(5.2) σ(H(t)) ≡ σ(H), σ_{ess}(H) = σ_{+} ∪ σ_{−}, σ_{p}(H) = {λ_{k}}_{k=1}^{p} ⊆ ℝ\σ_{ess}(H),

where p ∈ ℕ is finite. We introduce the sets

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

where σ is the (absolutely) continuous spectrum of H(t) and σ_{+}^{(1)} ∪ σ_{−}^{(1)}, σ^{(2)} are the parts which are of multiplicity one, two, respectively.

The Jost solutions ψ_{±}(z, n, t) are normalized such that

(5.4)  ψ_{±}(z, n, t) = ψ_{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

(5.5)  T_{∓}(λ, t)ψ_{±}(λ, n, t) = \overline{ψ_{∓}(λ, n, t)} + R_{∓}(λ, t)ψ_{∓}(λ, n, t),  λ ∈ σ_{∓},

which implies

(5.6) T_{±}(λ, t) := \frac{W (\overline{ψ_{±}(λ, t)}, ψ_{±}(λ, t))}{W (ψ_{∓}(λ, t), ψ_{±}(λ, t))}, R_{±}(λ, t) := −\frac{W (ψ_{∓}(λ, t), \overline{ψ_{±}(λ, t)})}{W (ψ_{∓}(λ, t), ψ_{±}(λ, t))},

λ ∈ σ_{±}. Here W_{n}(f, g) = a(n)(f(n)g(n + 1) − f(n + 1)g(n)) denotes the usual Wronski determinant.

To define the norming constants we need to remove the poles of ψ^{±}(z, n, t) by introducing

(5.7) ψ̃^{±}(z, n, t) = δ_{±}(z, t)ψ^{±}(z, n, t),  δ_{±}(z, t) := ∏_{μ_{j}^{±}(t)∈M_{±}(t)}(z − μ_{j}^{±}(t)),

where

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

The norming constants γ_{±,k}(t) corresponding to λ_{k} ∈ σ_{p}(H) are then given by

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

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

(5.10)  ψ̂_{±}(z, n, t) = exp(α_{r}^{±}(z, t))ψ_{±}(z, n, t)

satisfy

(5.11) H(t)ψ̂_{±}(z, n, t) = zψ̂_{±}(z, n, t),  \frac{d}{dt}ψ̂_{±}(z, n, t) = P_{2r+2}(t)ψ̂_{±}(z, n, t).

Proof. We proceed as in [31, 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 [7, Theorem 4.2]), and the derivatives are equal to the derivatives of the Baker– Akhiezer functions as n → ±∞.

For z ∈ ℂ\σ, the solution u_{±}(z, n, t) of (5.11) with initial condition ψ_{±}(z, n, 0) ∈ ℓ_{±}^{2}(ℤ) remains square summable near ±∞ for all t ∈ ℝ (see [32] or [33, Lemma 12.16]), that is, u_{±}(z, n, t) = C_{±}(z, t)ψ_{±}(z, n, t). Letting n → ±∞ we see C_{±}(z, t) = 1. The general result for all z ∈ ℂ now follows from continuity. □

This implies

Theorem 5.2. 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) exp(α_{r}^{∓}(λ, t) − \overline{α_{r}^{±}(λ, t)}),
R_{±}(λ, t) = R_{±}(λ, 0) exp(α_{r}^{±}(λ, t) − \overline{α_{r}^{±}(λ, t)}),
γ_{±,k}(t) = γ_{±,k}(0)\frac{δ_{±}^{2}(λ_{k}, 0)}{δ_{±}^{2}(λ_{k}, t)} exp(2α_{r}^{±}(λ_{k}, t)),  1 ≤ k ≤ p.

(5.12)

Proof. The Wronskian of two solutions satisfying (5.11) does not depend on n or t (see [32], [33, Lemma 12.15]), hence

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

The time dependence of γ_{±,k}(t) follows from ∥U_{r}(t, 0)ψ̃_{±}(λ_{k}, ., 0)∥ = ∥ψ̃_{±}(λ_{k}, ., 0)∥.

□

Remark 5.3. Note that we have (5.13)

exp (α_{r}^{±}(λ, t) − \overline{α_{r}^{±}(λ, t)}) = exp (P_{±}(λ) ∫_{0}^{t} \frac{G_{q,r}^{±}(λ, 0, s)}{∏_{j=1}^{g_{±}}(λ − μ_{j}^{±}(s))}ds) , λ ∈ σ_{±},

where μ_{j}^{±}(t) are the Dirichlet eigenvalues. Moreover, in case of the Toda lattice, where r = 0, we have G_{q,0}^{±}(λ, n, t) = 1 and H_{q,1}^{±}(λ, n, t) = λ − b_{q}^{±}(n, t).

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_{±,k}(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 [10]. More precisely, one needs to solve the GLM equation

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

for K_{±}(n, m, t), where according to Theorem 5.2 the kernel F_{±}(m, n, t) is given by

Theorem 5.4. The time dependence of the kernel of the Gel’fand–Levitan– Marchenko equation is given by

F_{±}(m, n, t) = \frac{1}{π} Re ∫_{σ_{±}} R_{±}(λ, 0)ψ̂_{q}^{±}(λ, m, t)ψ̂_{q}^{±}(λ, n, t)\frac{∏_{j=1}^{g_{±}}(λ − μ_{j}^{±}(0))}{P_{±}(λ)}dλ

+ \frac{1}{2πi} ∫_{σ_{∓}^{(1)}} |T_{∓}(λ, 0)|^{2}ψ̂_{q}^{±}(λ, m, t)ψ̂_{q}^{±}(λ, n, t)\frac{∏_{j=1}^{g_{±}}(λ − μ_{j}^{±}(0))}{P_{∓}(λ)}dλ (5.15)

+ ∑_{k=1}^{p} γ_{±,k}(0)ψ̆_{q}^{±}(λ_{k}, m, t)ψ̆_{q}^{±}(λ_{k}, n, t),

where ψ̆_{q}^{±}(z, m, t) = δ_{±}(z, 0)ψ̂_{q}^{±}(z, m, t).

Proof. The kernel F_{±}(m, n, 0) is derived in [10, Theorem 4.1]. Observe that α_{r}^{∓}(λ, t) are real valued on the set σ_{∓}^{(1)} and

exp (α_{r}^{±}(λ, t) + \overline{α_{r}^{±}(λ, t)}) = ∏_{j=1}^{g_{±}} \frac{λ − μ_{j}^{±}(t)}{λ − μ_{j}^{±}(0)}, λ ∈ σ_{±},

then our result follows from (5.10) and Theorem 5.2. □

By [10] this equation is uniquely solvable and the solution of the Toda hierarchy can be obtained from either K_{+}(n, m, t) or K_{−}(n, m, t) by virtue of

a(n, t) = a_{q}^{+}(n, t)\frac{K_{+}(n + 1, n + 1, t)}{K_{+}(n, n, t)} = a_{q}^{−}(n, t)\frac{K_{−}(n, n, t)}{K_{−}(n + 1, n + 1, t)},
b(n, t) = b_{q}^{+}(n, t) + a_{q}^{+}(n, t)\frac{K_{+}(n, n + 1, t)}{K_{+}(n, n, t)} − a_{q}^{+}(n − 1, t)\frac{K_{+}(n − 1, n, t)}{K_{+}(n − 1, n − 1, t)},

= b_{q}^{−}(n, t) + a_{q}^{−}(n − 1, t)\frac{K_{−}(n, n − 1, t)}{K_{−}(n, n, t)} − a_{q}^{−}(n, t)\frac{K_{−}(n + 1, n, t)}{K_{−}(n + 1, n + 1, t)}. (5.16)

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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/