SCATTERING THEORY FOR JACOBI OPERATORS WITH STEPLIKE QUASI-PERIODIC BACKGROUND

IRYNA EGOROVA, JOHANNA MICHOR, AND GERALD TESCHL

Abstract. We develop direct and inverse scattering theory for Jacobi operators with steplike quasi-periodic finite-gap background in the same isospectral class. We derive the corresponding Gel’fand-Levitan-Marchenko equation and find minimal scattering data which determine the perturbed operator uniquely. In addition, we show how the transmission coefficients can be reconstructed from the eigenvalues and one of the reflection coefficients.

1. Introduction

In this paper we consider direct and inverse scattering theory for Jacobi operators with steplike quasi-periodic finite-gap background in the same isospectral class using the Marchenko [16] approach.

Scattering theory for Jacobi operators with a constant background is a classical topic first developed on an informal level by Case in [4]. The first rigorous results were established by Guseinov [12] with further extensions by Teschl [19], [20]. The case of a quasi-periodic finite-gap background was recently investigated by us in [8] (see also [23] respectively [9], [17] for applications to the Toda hierarchy).

Our motivation for the investigation of a steplike situation is twofold. First of all, steplike potentials are a simple model in quantum mechanics which have attracted renewed interest due to their possible applications in mesoscopic solid state structures. We refer for example to [6] where the discrete one dimensional Schrödinger equation is used as a simple one-band tight binding model to explain some of the essential qualitative properties of the Wannier-Stark ladders. The interested reader should also consult [11] or [18] and the references therein for further information.

Our second motivation is the study of solitons on (quasi-)periodic backgrounds. While solitons on quasi-periodic backgrounds are well investigated objects, not much about their stability is known. In fact, as pointed out only recently in [13], the general believe that the stability problem for solitons on quasi-periodic backgrounds is similar to the one for solitons on constant background is wrong (for a detailed analysis using Riemann-Hilbert techniques see [14]). This is related to the fact that solitons on quasi-periodic backgrounds give rise to different spatial asymptotics which naturally leads to the type of operators studied here. Hence our results form the basis for an investigation of solitons on quasi-periodic backgrounds via the inverse scattering transform. For further details we refer to [10].

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

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

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

Though the case with different but constant background is well understood by now (see [7] and [3], [5], [22] for applications to the Toda lattice), the case with different (quasi-)periodic background still is in its infancy. First results for the case of two period two operators with a special choice for the respective spectra have been obtained in [1]. However, to the best of our knowledge the general problem is still open (even for the case of one-dimensional Schrödinger operators – see [11] and the references therein for a recent account). Our aim is to fill this gap for the case of isospectral background operators.

After recalling some necessary facts on algebro-geometric quasi-periodic finitegap operators in Section 2, we construct the transformation operators and investigate the properties of the scattering data in Section 3. In particular, we show how both transmission coefficients can be reconstructed from the eigenvalues and the corresponding reflection coefficients. 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.

2. Quasi-periodic finite-gap operators

As a preparation for our next section we first need to recall some facts on quasiperiodic finite-gap Jacobi operators which contain all periodic operators as a special case. We refer to [20, Chapter 9].

Let 𝕄 be the Riemann surface associated with the following function

(2.1)  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.

Let {a_{j}, b_{j}}_{j=1}^{g} be loops on the Riemann surface 𝕄 representing the canonical generators of the fundamental group π_{1}(𝕄). We require a_{j} to surround the points E_{2j−1}, E_{2j} (thereby changing sheets twice) and b_{j} to surround E_{0}, E_{2j−1} counterclockwise on the upper sheet, with pairwise intersection indices given by

(2.2)  a_{j} ∘ a_{k} = b_{j} ∘ b_{k} = 0,  a_{j} ∘ b_{k} = δ_{jk},  1 ≤ j, k ≤ g.

The corresponding canonical basis {ζ_{j}}_{j=1}^{g} for the space of holomorphic differentials can be constructed by

(2.3)  \underline{ζ} = ∑_{j=1}^{g} \underline{c}(j)\frac{π^{j−1}dπ}{R_{2g+2}^{1/2}} ,

where the constants \underline{c}(.) are given by

c_{j}(k) = C_{jk}^{−1}, C_{jk} = ∫_{a_{k}} \frac{π^{j−1}dπ}{R_{2g+2}^{1/2}} = 2 ∫_{E_{2k−1}}^{E_{2k}} \frac{z^{j−1}dz}{R_{2g+2}^{1/2}(z)} ∈ ℝ.

The differentials fulfill

(2.4)  ∫_{a_{j}} ζ_{k} = δ_{j,k},  ∫_{b_{j}} ζ_{k} = τ_{j,k},  τ_{j,k} = τ_{k,j},  1 ≤ j, k ≤ g.

Finally we will need ω_{pq}, the normalized Abelian differential of the third kind with poles at p and q, that is, ω_{pq} has vanishing a-periods and first order poles with residues +1, −1 at p, q, respectively.

Now pick g numbers (the Dirichlet eigenvalues)

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

where \underline{Ξ}_{p_{0}} is the vector of Riemann constants 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)) has precisely g zeros μ̂_{j}(n) (with μ̂_{j}(0) = μ̂_{j}), where θ(\underline{z}) is the Riemann theta function of 𝕄.

With this notation our quasi-periodic finite-gap operator is given by

(2.7)  H_{q}f(n) = a_{q}(n)f(n + 1) + a_{q}(n − 1)f(n − 1) + b_{q}(n)f(n),

where

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

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

The constants ã, b̃, c_{j}(g) depend only on the Riemann surface (see [20, Section 9.2]). Introduce the Baker-Akhiezer function

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

where C(n, 0) is real-valued,

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

and ω_{∞_{+},∞_{−}} is the Abelian differential of the third kind with poles at ∞_{+} respectively ∞_{−}. The two branches ψ_{q,±}(z, n) = ψ_{q}(p, n), p = (z, ±), of the Baker-Akhiezer function are solutions (in the weak sense) of

H_{q}ψ_{q,±}(z, n) = zψ_{q,±}(z, n)

which are linearly independent away from the branch points. In fact, their Wronskian is given by

W_{q}(ψ_{q,−}(z), ψ_{q,+}(z)) = a_{q}(n) (ψ_{q,−}(z, n), ψ_{q,+}(z, n + 1) − ψ_{q,−}(z, n + 1), ψ_{q,+}(z, n))

= \frac{R_{2g+2}^{1/2}(z)}{∏_{j=1}^{g}(z − μ_{j})}. (2.10)

It is well-known that the spectrum of H_{q} consists of g + 1 bands

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

Note that ψ_{q,±}(z, n) are discontinuous along the spectrum and we set ψ_{q,±}(λ, n) = lim_{ε↓0} ψ_{q,±}(λ + iε, n), λ ∈ σ(H_{q}). This implies lim_{ε↓0} ψ_{q,±}(λ − iε, n) = \overline{ψ_{q,±}(λ, n)} = ψ_{q,∓}(λ, n), λ ∈ σ(H_{q}). For further information and proofs we refer to [20, Chapter 9].

3. Scattering data

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

(3.1)  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 2 to distinguish between the corresponding data of H_{q}^{+} and H_{q}^{−}. To avoid excessive sub/superscripts we abbreviate

(3.2)  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)}.

In addition, we split the set of Dirichlet eigenvalues into three parts:

M^{±} = {μ_{j}^{±} | μ_{j}^{±} ∈ ℝ\Σ is a pole of ψ_{q}^{±}(z, 1)},
M̄^{±} = {μ_{j}^{±} | μ_{j}^{±} ∈ ℝ\Σ is a pole of ψ̄_{q}^{±}(z, 1)},
(3.3)  M̃^{±} = {μ_{j}^{±} | μ_{j}^{±} ∈ ∂Σ},

and introduce

(3.4)  ψ̂_{q}^{±}(z, n) = ( ∏_{μ∈M^{±}}(z − μ))ψ_{q}^{±}(z, n),

which are holomorphic for all z ∈ ℂ\Σ.

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

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

and denote the corresponding operator by H.

The special case H_{q}^{−} = H_{q}^{+} has been exhaustively studied in [8] (see also [23]) and several results are straightforward generalizations. In such situations we will simply refer to [8] and only point out possible differences. In fact, we begin by recalling two theorems from [8]:

Theorem 3.1. Assume (3.5). Then there exist weak solutions ψ_{±}(z, .) of Hψ = zψ satisfying

(3.6)  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. Moreover, ψ_{±}(z, .) are continuous (resp. holomorphic) with respect to z whenever ψ_{q}^{±}(z, .) are and (3.7)

ψ_{±}(z, n) = \frac{z^{∓n}}{A_{±}(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

A_{+}(n) = ∏_{j=n}^{∞} \frac{a(j)}{a_{q}^{+}(j)},  B_{+}(n) = ∑_{m=n+1}^{∞}(b_{q}^{+}(m) − b(m)),

(3.8)  A_{−}(n) = ∏_{j=−∞}^{n−1} \frac{a(j)}{a_{q}^{−}(j)},  B_{−}(n) = ∑_{m=−∞}^{n−1}(b_{q}^{−}(m) − b(m)).

Defining ψ̂_{±}(z, n) using ψ̂_{q}^{±}(z, n) (instead of ψ_{q}^{±}(z, n)), we have that ψ̂_{±}(z, n) are holomorphic in ℂ\Σ.

Theorem 3.2. Assume (3.5). 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.

It will be convenient to identify ℂ\Σ with Π_{+} and regard the functions ψ_{±}, ψ_{q}^{±} as functions on Π_{+}. Furthermore, we extend them by continuity to ∂Π_{+}, that is, ψ_{+}((λ, ±), n) = lim_{ε↓0} ψ_{+}(λ ± iε, n), etc, for λ ∈ Σ. Consequently we have ψ_{q}^{±}(p^{∗}, n) = \overline{ψ_{q}^{±}(p, n)} and ψ_{±}(p^{∗}, n) = \overline{ψ_{±}(p, n)} for p ∈ ∂Π_{+}.

Note: Unfortunately this disagrees with our previous definition of ψ_{q}^{−} which was originally defined as the branch on Π_{−}. Moreover, one should emphasize that ψ_{±}(p, n) does not have a continuation to Π_{−} in general.

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

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

we can define

K_{±}(n, m) = ∫_{∂Π_{+}} ψ_{±}(p, n)\overline{ψ_{q}^{±}(p, m)}dω^{±}

= 2Re ∫_{Σ} ψ_{±}(λ, n)\overline{ψ_{q}^{±}(λ, m)}dω^{±}. (3.10)

Then K_{±}(n, m) satisfy ([8]):

Lemma 3.3. Assume (3.5). The Jost solutions ψ_{±}(z, n) can be represented as

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

where the kernels K_{±}(n, .) satisfy K_{±}(n, m) = 0 for ±m < ±n and (3.12)

|K_{±}(n, m)| ≤ C_{±}(n) ∑_{j=[ \frac{m+n}{2} ]±1}^{±∞} (|a(j) − a_{q}^{±}(j)| + |b(j) − b_{q}^{±}(j)|), ±m > ±n > 0.

The functions C_{±}(n) > 0 decrease as n → ±∞ and depend only on H_{q}^{±} and the value of the sums in (3.5).

Associated with K_{±}(n, m) is the operator

(3.13)  (K_{±}f)(n) = ∑_{m=n}^{±∞} K_{±}(n, m)f(m),  f ∈ ℓ_{±}^{∞}(ℤ, ℂ),

which acts as a transformation operator for the pair τ, τ_{q}^{±}.

Theorem 3.4. Let τ_{q}^{±} and τ be the quasi-periodic and perturbed Jacobi difference expressions, respectively. Then

(3.14)  τK_{±}f = K_{±}τ_{q}^{±}f,  f ∈ ℓ_{±}^{∞}(ℤ, ℂ).

Furthermore, for n ∈ ℤ we have

(3.15)  \frac{a(n)}{a_{q}^{+}(n)} = \frac{K_{+}(n + 1, n + 1)}{K_{+}(n, n)}
\frac{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)}.

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

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

where

(3.17)  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

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

Using

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

we see that T_{±}(z) admit a meromorphic extension to ℂ\Σ. Since W (ψ̂_{−}(z), ψ̂_{+}(z)) is holomorphic in ℂ\Σ with simple zeros at the eigenvalues ρ_{k} (see [20, Section 2.2]) we obtain the following behavior:

(3.20)  T_{±}(z) = \frac{∏_{μ_{j}^{∓}∈M^{∓}}(z − μ_{j}^{∓})}{∏_{μ_{j}^{±}∈M̄^{±}}(z − μ_{j}^{±})} \frac{D_{±}(z)}{∏_{k=1}^{q}(z − ρ_{k})},

where D_{±}(z) are holomorphic and do not vanish in ℂ\Σ. That is, in general the transmission coefficients have simple poles at the eigenvalues ρ_{j} of H. In addition, there are simple poles at μ_{j}^{±} ∈ M̄^{±} and simple zeros at μ_{j}^{∓} ∈ M^{∓}. A pole at μ_{j}^{±} could cancel with a zero at μ_{j}^{∓} or could give a second order pole if μ_{j}^{±} = ρ_{k}.

The Plücker identity (c.f. [20, (2.169)]) implies

(3.21)  α_{+}(λ)\overline{α_{−}(λ)} = 1 − β_{+}(λ)β_{−}(λ),

and using

(3.22)  α_{+}(λ) = α_{−}(λ) ∏_{j=1}^{g} \frac{λ − μ_{j}^{+}}{λ − μ_{j}^{−}} ,  \overline{β_{+}}(λ) = −β_{−}(λ) ∏_{j=1}^{g} \frac{λ − μ_{j}^{+}}{λ − μ_{j}^{−}} ,

we obtain

(3.23)  |α_{±}(λ)|^{2} = ∏_{j=1}^{g} \frac{λ − μ_{j}^{±}}{λ − μ_{j}^{∓}} + |β_{±}(λ)|^{2}.

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

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

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

Lemma 3.5. The coefficients T_{±}(λ), R_{±}(λ) are continuous for λ ∈ Σ except at possibly the band edges E_{j}, and fulfill

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

In particular,

(3.27)  |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}, where

(3.28)

lim_{z→E} R_{2g+2}^{1/2}(z)\frac{R_{±}(z)+1}{T_{±}(z)} = 0,  E ≠ \{μ_{j}^{+}, μ_{j}^{−}\},
lim_{z→E} R_{2g+2}(z)\frac{R_{±}(z)+1}{T_{±}(z)} = 0,  E = μ_{j}^{∓} ≠ μ_{j}^{±},
lim_{z→E} \frac{R_{±}(z)−1}{T_{±}(z)} = 0,  E = μ_{j}^{±} ≠ μ_{j}^{∓},
lim_{z→E} R_{2g+2}^{1/2}(z)\frac{R_{±}(z)−1}{T_{±}(z)} = 0,  E = μ_{j}^{+} = μ_{j}^{−}.

The transmission coefficients T_{±}(λ) have a meromorphic continuation with

(3.29)  Res_{z=ρ_{k}} \frac{∏_{μ_{j}^{∓}∈M^{∓}}(z − μ_{j}^{∓})}{∏_{μ_{j}^{±}∈M̄^{±}∪M̃^{±}}(z − μ_{j}^{±})}T_{±}(z) = −\sqrt{γ_{+,k}γ_{−,k}}R_{2g+2}^{1/2}(ρ_{k}).

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

(3.30)  T_{±}(∞) = \frac{1}{K_{+}(0, 0)K_{−}(0, 0)}.

Proof. To prove (3.28), recall first that if E ∉ M̃^{±}, then ψ_{±}(λ, n) − \overline{ψ_{±}(λ, n)} → 0 as λ → E, and ψ_{±}(E, n) are bounded. If E ∈ M̃^{±}, then R_{2g+2}^{1/2}(λ)ψ_{±}(λ, n) and ψ_{±}(λ, n) + \overline{ψ_{±}(λ, n)} are bounded at E.

Thus using definition (3.17), we have as E ∉ M̃^{−} ∪ M̃^{+}

R_{2g+2}^{1/2}(λ)\frac{R_{±}(λ) + 1}{T_{±}(λ)} = ∏_{j=1}^{g}(λ − μ_{j}^{±})(W (ψ_{−}(λ), ψ_{+}(λ)) ∓ W (ψ_{∓}(λ), \overline{ψ_{±}(λ)})) → 0.

If E ∈ M̃^{±}\M̃^{∓}, then

\frac{R_{±}(λ) − 1}{T_{±}(λ)} = ±\frac{∏_{j=1}^{g}(λ − μ_{j}^{±})}{R_{2g+2}^{1/2}(λ)}W (ψ_{±}(λ) + \overline{ψ_{±}(λ)}, ψ_{∓}(λ)) → 0.

In the same manner one proves the remaining two equalities in (3.28). By [20, (2.33)],

(3.31)  W′(ψ̂_{−}(z), ψ̂_{+}(z))|_{z=ρ_{j}} = \frac{−1}{\sqrt{γ_{−,j}γ_{+,j}}} ,

which proves (3.29). Finally, (3.30) follows from (3.7). □

Observe that (3.25) implies |R_{−}(λ)| = |R_{+}(λ)|.

The sets

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

are called left/right scattering data for H.

Theorem 3.6. The transmission coefficients can be reconstructed from one reflection coefficient and the eigenvalues via

T_{+}(z) =\sqrt{\frac{θ(\underline{z}^{+}(∞_{+}, 0))θ(\underline{z}^{+}(∞_{−}, 0))}{θ(\underline{z}^{−}(∞_{+}, 0))θ(\underline{z}^{−}(∞_{−}, 0))}} \frac{θ(\underline{z}^{−}(ẑ^{∗}, 0))}{θ(\underline{z}^{+}(ẑ^{∗}, 0))}×

(3.33) × (∏_{j=1}^{q} exp (− ∫ ρ̂_{j} ; E(ρ_{j})ωẑ_{}_{ẑ^{∗}})) exp (\frac{1}{2πi} ∫_{Σ} ln(1 − |R_{±}|^{2})ωẑ_{}_{ẑ^{∗}}) ,

T_{−}(z) =\sqrt{\frac{θ(\underline{z}^{−}(∞_{+}, 0))θ(\underline{z}^{−}(∞_{−}, 0))}{θ(\underline{z}^{+}(∞_{+}, 0))θ(\underline{z}^{+}(∞_{−}, 0))}} \frac{θ(\underline{z}^{+}(ẑ, 0))}{θ(\underline{z}^{−}(ẑ, 0))}×

(3.34) × (∏_{j=1}^{q} exp (− ∫ ρ̂_{j} ; E(ρ_{j})ωẑ_{}_{ẑ^{∗}})) exp (\frac{1}{2πi} ∫_{Σ} ln(1 − |R_{±}|^{2})ωẑ_{}_{ẑ^{∗}}) ,

where ẑ = (z, +), the integral over Σ is taken on the upper sheet, and E(ρ) is E_{0} if ρ < E_{0}, either E_{2j−1} or E_{2j} if ρ ∈ (E_{2j−1}, E_{2j}), 1 ≤ j ≤ g, and E_{2g+1} if ρ > E_{2g+1}.

Furthermore, the phase shift between H_{q}^{+} and H_{q}^{−} can also be computed from one reflection coefficient and the eigenvalues

(3.35)  \underline{α}_{p_{0}}(D\underline{μ̂}^{+}) − \underline{α}_{p_{0}}(D\underline{μ̂}^{−}) = ∑_{j=1}^{q} ∫ ρ̂_{j} ; ρ̂_{j}^{∗} \underline{ζ} − \frac{1}{2πi} ∫_{∂Π_{+}}ln(1 − |R_{±}|^{2})\underline{ζ}.

Proof. First of all note that Im ∫ ω_{pp^{∗}} is the Green’s function of Π_{+} and that

(3.36)  B(z, ρ) = exp (∫ ρ̂ ; E(ρ)ωẑ_{}_{ẑ^{∗}})

is the Blaschke factor (see [21]). That is, B(z, ρ) is a multivalued holomorphic function, which vanishes at z = ρ and satisfies |B(λ, ρ)| = 1 for λ ∈ Σ.

We start by considering the multivalued function

(3.37) t_{+}(z) = \sqrt{\frac{θ(\underline{z}^{−}(∞_{+}, 0))θ(\underline{z}^{−}(∞_{−}, 0))}{θ(\underline{z}^{+}(∞_{+}, 0))θ(\underline{z}^{+}(∞_{−}, 0))}} \frac{θ(\underline{z}^{+}(ẑ^{∗}, 0))}{θ(\underline{z}^{−}(ẑ^{∗}, 0))} (∏_{j=1}^{q} B(z, ρ_{j})) T_{+}(z)

which has neither zeros nor poles on Π_{+}. Using the same argument as in [20, eq. (9.27)] one verifies that θ(\underline{z}^{+}(∞_{±}, 0))/θ(\underline{z}^{−}(∞_{±}, 0)) is positive. The absolute value of t_{+}(z) is single-valued and, using \overline{\underline{z}^{±}(λ̂, n)} = \underline{z}^{±}(λ̂^{∗}, n) mod ℤ^{g} for λ̂ = (λ, +) with λ ∈ Σ, one obtains

|t_{+}(λ)|^{2} = \frac{θ(\underline{z}^{−}(∞_{+}, 0))θ(\underline{z}^{−}(∞_{−}, 0))}{θ(\underline{z}^{+}(∞_{+}, 0))θ(\underline{z}^{+}(∞_{−}, 0))} \frac{θ(\underline{z}^{+}(λ̂^{∗}, 0))}{θ(\underline{z}^{−}(λ̂^{∗}, 0))} \frac{θ(\underline{z}^{+}(λ̂, 0))}{θ(\underline{z}^{−}(λ̂, 0))}|T_{+}(λ)|^{2}

= \frac{∏(λ − μ_{j}^{+})}{∏(λ − μ_{j}^{−})}|T_{+}(λ)|^{2}. (3.38)

Here the last equality follows since the ratio of theta functions extends to a (singlevalued) meromorphic function on ℂ which is one at ∞. Hence |t_{+}(λ)|^{2} = 1 − |R_{±}(λ)|^{2} for λ ∈ Σ by (3.27) and ln |t_{+}(z)| can be reconstructed from its boundary values using Green’s function:

(3.39)  |t_{+}(z)| = exp (Re\frac{1}{2πi} ∫_{Σ} ln(1 − |R_{±}|^{2})ωẑ_{}_{ẑ^{∗}}) .

So (3.33) holds at least when taking absolute values. But since both sides are meromorphic, they can only differ by a factor of absolute value one. Since both sides are positive at ∞_{+}, they are equal. The claim for T_{−}(z) follows analogously.

To show (3.35) we use that, since T_{±} are single-valued, there is no jump when we go around b-cycles. Hence the jump when going around b_{ℓ} of all factors must add up to zero, which is just (3.35). □

Combining this result with (3.29) we obtain:

Corollary 3.7. One of the scattering data S_{−}(H) or S_{+}(H) determines the other.

4. The Gel’fand-Levitan-Marchenko equation

Finally we want to derive the Gel’fand-Levitan-Marchenko equation and show that one of the sets S_{−}(H) or S_{+}(H) uniquely determines H.

Theorem 4.1. The kernel K_{±}(n, m) of the transformation operator satisfies the Gel’fand-Levitan-Marchenko equation

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

+ ∑_{j=1}^{q} γ_{±,j}ψ̂_{q}^{±}(ρ_{j}, n)ψ̂_{q}^{±}(ρ_{j}, m). (4.2)

Proof. As already done for ψ_{±}, we regard T_{±}(z) as functions on Π_{+} and R_{±}(λ) as functions on ∂Π_{+} by setting R_{±}((λ, +)) = R_{±}(λ) respectively R_{±}((λ, −)) = \overline{R_{±}(λ)}.

Computing the Fourier coefficients of

(4.3)  T_{±}(λ)ψ_{∓}(λ, n) = R_{±}(λ)ψ_{±}(λ, n) + \overline{ψ_{±}(λ, n)}

as in [8, Sec. 7] one obtains by (3.7), (3.9), (3.19), (3.30), (3.31), ψ̂_{−}(ρ_{j}, n) = c_{j}^{−}ψ̂_{+}(ρ_{j}, n), and the residue theorem

∫_{∂Π_{+}} T_{+}(p)ψ_{−}(p, n)ψ_{q}^{+}(p, m)dω^{+} = ∫_{∂Π_{+}} \frac{ψ_{−}(p, n)ψ_{q}^{+}(p, m)}{W (ψ_{−}(p), ψ_{+}(p))}dπ

= \frac{δ(n, m)}{K_{+}(n, n)} + ∑_{j=1}^{q} Res_{ρ_{j}}(\frac{ψ̂_{−}(p, n)ψ̂_{q}^{+}(p, m)}{W (ψ̂_{−}(p), ψ̂_{+}(p))})

= \frac{δ(n, m)}{K_{+}(n, n)} − ∑_{j=1}^{q} γ_{+,j}ψ̂_{+}(ρ_{j}, n)ψ̂_{q}^{+}(ρ_{j}, m).

The right hand side follows analogous to [8]. □

Note that while the scattering data depend on the particular normalization chosen for ψ_{q}^{±}, the kernel of Gel’fand-Levitan-Marchenko equation is of course independent of this normalization.

Theorem 4.2. For n ∈ ℤ, the Gel’fand-Levitan-Marchenko operator

(4.4)  F_{n}^{±} : ℓ^{2} → ℓ^{2},  F_{n}^{±}f(j) = ∑_{l=0}^{∞} F^{±}(n ± l, n ± j)f(l),

is Hilbert-Schmidt. Moreover, 1 + F_{n}^{±} is positive and hence invertible. In particular, the Gel’fand-Levitan-Marchenko equation (4.1)

(4.5)  (1 + F_{n}^{±})K_{±}(n, n ± .) = (K_{±}(n, n))^{−1}δ_{0}

has a unique solution and S_{+}(H) or S_{−}(H) uniquely determine H.

Proof. The proof can be done as in [8, Theorem 7.5], since by equation (3.27) we have |R_{±}(λ)| < 1 for λ ∈ Σ\∂Σ. □

To finish the direct scattering step we summarize the properties of S_{±}(H).

Hypothesis H.4.3. The scattering data

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

satisfy the following conditions:

(i) The reflection coefficients R_{±}(λ) are continuous except possibly at the band edges E and fulfill |R_{±}(λ)| < 1 for λ ∈ Σ\∂Σ.

The Fourier coefficients

(4.6)  F̃^{±}(l, m) = 2Re ∫_{Σ} R_{±}(λ)ψ_{q}^{±}(λ, l)ψ_{q}^{±}(λ, m)dω^{±}

satisfy

|F̃^{±}(n, m)| ≤ ∑_{j=n+m}^{±∞} q(j),  q(j) ≥ 0,  |j|q(j) ∈ ℓ^{1}(ℤ),

∑_{n=n_{0}}^{±∞} |n||F̃^{±}(n, n) − F̃^{±}(n ± 1, n ± 1)| < ∞,

∑_{n=n_{0}}^{±∞} |n||a_{q}^{±}(n)F̃^{±}(n, n + 1) − a_{q}^{±}(n − 1)F̃^{±}(n − 1, n)| < ∞.

(ii) The values ρ_{j} ∈ ℝ\Σ are distinct and γ_{±,j} ≥ 0 for 1 ≤ j ≤ q.

(iii) ln(1 − |R_{±}|^{2}) is integrable on Σ and T_{±}(λ) defined via (3.33), (3.34) extend to single-valued functions on Π_{+} with

(4.7)  T_{−}(λ) = T_{+}(λ) ∏_{j=1}^{g} \frac{λ − μ_{j}^{+}}{λ − μ_{j}^{−}} .

(iv) Transmission and reflection coefficients satisfy (3.28), (3.30), and the consistency conditions

\frac{R_{∓}(λ)}{\overline{R_{±}(λ)}} = −\frac{T_{±}(λ)}{\overline{T_{±}(λ)}}, γ_{+,j} γ_{−,j} = \frac{(Res_{ρ_{j}} \frac{∏_{μ{∓ \atop j}∈M^{∓}} (z−μ_{j}^{∓})}{∏_{μ{± \atop j}∈M̄^{±}∪M̃^{±}} (z−μ_{j}^{±})}T_{±}(z))^{2}}{R_{2g+2}(ρ_{j})} .

5. Inverse scattering theory

In this section we reconstruct the operator H from a given set S_{+} or S_{−} and given quasi-periodic Jacobi operators H_{q}^{±}.

If S_{±} (satisfying H.4.3 (i)–(ii)) and H_{q}^{±} are known, we can construct F^{±}(l, m) via formula (4.2) and thus derive the Gel’fand-Levitan-Marchenko equation, which has a unique solution by Theorem 4.2. We obtain that

K_{±}(n, n) = ⟨δ_{0}, (1 + F_{n}^{±})^{−1}δ_{0}⟩^{1/2}
(5.1)  K_{±}(n, n ± j) = \frac{1}{K_{±}(n, n)}⟨δ_{j}, (1 + F_{n}^{±})^{−1}δ_{0}⟩.

Since 1 + F_{n}^{±} is positive, K_{±}(n, n) is positive and we can set (see Theorem 3.4)

(5.2)  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)}.

Let H_{+}, H_{−} be the associated Jacobi operators. As in [8] one proves

Lemma 5.1. Suppose a given set S_{±} satisfies H.4.3 (i)–(ii). Then the sequences defined in (5.2) satisfy n|a_{±}(n) − a_{q}^{±}(n)|, n|b_{±}(n) − b_{q}^{±}(n)| ∈ ℓ_{±}^{1}(ℕ).

Moreover, ψ_{±}(λ, n) = ∑_{m=n}^{±∞} K_{±}(n, m)ψ_{q}^{±}(λ, m), where K_{±}(n, m) is the solution of the Gel’fand-Levitan-Marchenko equation, satisfies τ_{±}ψ_{±} = λψ_{±}.

We set

(5.3)  W (λ) = \frac{R_{2g+2}^{1/2}(λ)}{T_{±}(λ) ∏_{j=1}^{g}(λ − μ_{j}^{±})}.

According to H.4.3 (iii), this function is holomorphic in ℂ\(M^{±} ∪ M̄^{±} ∪ ∂Σ). Now we can prove the main result of this section.

Theorem 5.2. Hypothesis H.4.3 is necessary and sufficient for sets S_{±} to be the left/right scattering data of a unique Jacobi operator H associated with sequences a, b satisfying (3.5).

Proof. Necessity has been established in the previous section. By Lemma 5.1, we know existence of sequences a_{±}, b_{±} and corresponding solutions ψ_{±}(z, n) associated with S_{+} (or S_{−}). Hence it remains to establish a_{+}(n) = a_{−}(n) and b_{+}(n) = b_{−}(n).

We study the following part of the GLM-equation

(5.4)  ∑_{m∈ℤ} Φ_{+}(n, m)\overline{ψ_{q}^{+}(λ, m)},  Φ_{+}(n, .) := ∑_{l=n}^{∞} K_{+}(n, l)F̃^{+}(l, .) ∈ ℓ_{+}^{1}(ℤ)

and obtain as in [8, Theorem 8.2]

(5.5)  T_{+}(λ)h_{−}(λ, n) = \overline{ψ_{+}(λ, n)} + R_{+}(λ)ψ_{+}(λ, n),  λ ∈ Σ,

where

h_{−}(λ, n) = \frac{\overline{ψ_{q}^{+}(λ, n)}}{T_{+}(λ)}(\frac{1}{K_{+}(n, n)} + ∑_{m=−∞}^{n−1} Φ_{+}(n, m)\frac{\overline{ψ_{q}^{+}(λ, m)}}{\overline{ψ_{q}^{+}(λ, n)}}

+ ∑_{j=1}^{q} γ_{+,j}ψ̂_{+}(ρ_{j}, n)\frac{W_{n−1}(ψ̂_{q}^{+}(ρ_{j}), \overline{ψ_{q}^{+}(λ)})}{(λ − ρ_{j})\overline{ψ_{q}^{+}(λ, n)}}). (5.6)

Clearly, using ψ̄_{q}^{±}(λ, n) = \overline{ψ_{q}^{±}(λ, n)}, the function h_{+}(λ, n) extends to a meromorphic function in ℂ\Σ.

Similar we obtain a function h_{+}(z, n). The functions h_{∓}(z, n) have simple poles at μ_{j}^{∓} ∈ M^{∓} and the following limits

(5.7)  h_{∓}(z, n) = \frac{z^{±n}}{K_{±}(n, n)T_{±}(∞)}( ∏_{j=0}^{n−1} a_{q}^{±}(j))^{∓1}(1 + O(\frac{1}{z})), z → ∞,

(5.8) lim_{z→ρ_{j}} h_{∓}(z, n) = \frac{γ_{±,j}ψ̂_{±}(ρ_{j}, n)}{\sqrt{γ_{+,j}γ_{−,j}} ∏_{μ∈M^{∓}}(ρ_{j} − μ)}.

By virtue of the consistency condition T_{±}(λ)\overline{R_{±}(λ)} = −\overline{T_{±}(λ)}R_{∓}(λ) we obtain

(5.9)  \overline{h_{±}(λ, n)} + R_{±}(λ)h_{±}(λ, n) = ψ_{∓}(λ, n)T_{±}(λ),  λ ∈ Σ.

Eliminating R_{±}(λ) in (5.5) and (5.9) and using (5.3) yields

R_{2g+2}^{−1/2}(λ) ∏_{j=1}^{g}(λ − μ_{j}^{±})(\overline{h_{±}(λ, n)}ψ_{±}(λ, n) − \overline{ψ_{±}(λ, n)}h_{±}(λ, n))

= \frac{ψ_{+}(λ, n)ψ_{−}(λ, n) − h_{+}(λ, n)h_{−}(λ, n)}{W (λ)} = G(λ, n), λ ∈ Σ. (5.10)

Observe that G(λ, n) extends to a meromorphic function on ℂ\Σ which is continuous on the interior of Σ with equal real limits from above and below: G(λ±i0, n) = \overline{G(λ ∓ i0, n)} = G(λ ± i0, n). So by the Schwarz reflection principle G(z, n) is holomorphic on the set ℂ\(M^{±} ∪ M̄^{±} ∪ {ρ_{k}}_{k=1}^{q} ∪ ∂Σ).

Since the difference ψ_{+}ψ_{−}−h_{+}h_{−} vanishes at the points ρ_{k} by (5.8) and H.4.3 (iv), the poles there are removable.

Next let us investigate the behavior at the band edges. Since G(z, n) has at most poles, it is sufficient to control the behavior from one direction. To this end we use (3.28) and the identity (5.10). If E ∉ M̃^{+} ∪ M̃^{−},

lim_{λ→E} R_{2g+2}^{1/2}(λ) ∏_{j=1}^{g}(λ − μ_{j}^{±})h_{∓}(λ, n)\overline{ψ_{∓}(λ, n)}

= lim_{λ→E} \frac{R_{2g+2}^{1/2} ∏_{j=1}^{g}(λ − μ_{j}^{±})}{T_{±}}(\overline{ψ_{±}} + R_{±}ψ_{±})\overline{ψ_{∓}}

= lim_{λ→E} \frac{R_{2g+2}^{1/2} ∏_{j=1}^{g}(λ − μ_{j}^{±})}{T_{±}}((R_{±} + 1)ψ_{±} + \overline{ψ_{±}} − ψ_{±})\overline{ψ_{∓}} = 0.

For E ∈ M̃^{+} ∩ M̃^{−}, the functions ψ_{±}(λ, .) + \overline{ψ_{±}(λ, .)} are bounded in the vicinity of E ∈ M̃^{±},

lim_{λ→E} \frac{R_{2g+2}^{1/2} ∏_{j=1}^{g}(λ − μ_{j}^{±})}{T_{±}}(\overline{ψ_{±}} + R_{±}ψ_{±})\overline{ψ_{∓}}

= \frac{R_{2g+2}^{1/2} ∏_{j=1}^{g}(λ − μ_{j}^{±})}{T_{±}}(ψ_{±} + \overline{ψ_{±}})\overline{ψ_{∓}} + ∏_{j=1}^{g}(λ − μ_{j}^{±})ψ_{±}\overline{ψ_{±}} \frac{R_{2g+2}^{1/2}(R_{±} − 1)}{T_{±}} = 0.

Analogously, one proves that R_{2g+2}(z)G(z, n) vanishes at the points E ∈ M̃_{±}\M̃_{∓}. Therefore this function is holomorphic on ℂ and R_{2g+2}(z)G(z, n) = O(z − E) for E ∈ ∂Σ. Thus G(z, n) has only removable singularities and since G(z, n) → 0 as z → ∞, Liouville’s theorem implies G(z, n) ≡ 0 and

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

For z → ∞ we obtain by (3.7) and (5.7)

T_{+}(∞)T_{−}(∞) = (\frac{1}{K_{+}(0, 0)K_{−}(0, 0)} ∏_{j=0}^{n−1} \frac{a_{−}(j)}{a_{+}(j)})^{2}

and therefore by (3.30)

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

It remains to prove b_{+}(n) = b_{−}(n). Proceeding as for G(λ, n) we can show that

\frac{ψ_{+}(λ, n)ψ_{−}(λ, n + 1) − h_{+}(λ, n + 1)h_{−}(λ, n)}{W (λ)}

= \frac{∏_{j=1}^{g}(λ − μ_{j}^{+})}{R_{2g+2}^{1/2}(λ)}(\overline{h_{+}(λ, n + 1)}ψ_{+}(λ, n) − \overline{ψ_{+}(λ, n)}h_{+}(λ, n + 1)) (5.12)

is a constant equal to −1/a(n). Thus

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

Computing the asymptotics as z → ∞ (compare (3.7)) we see

(5.13)  0 = W̄ (z, n) − W̄ (z, n − 1) = \frac{b_{+}(n) − b_{−}(n)}{A_{+}(0)A_{−}(0)}

and in particular b_{+}(n) = b_{−}(n) ≡ b(n). □

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.

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Kharkiv National University, 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/