Trace Formulas for Jacobi Operators in Connection with Scattering Theory for Quasi-Periodic Background

Johanna Michor and Gerald Teschl

Abstract. We investigate trace formulas for Jacobi operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein’s spectral shift theory. In particular we establish the conserved quantities for the solutions of the Toda hierarchy in this class.

1. Introduction

Scattering theory for Jacobi operators H with periodic (respectively more general) background has attracted considerable interest recently. In [14] Volberg and Yuditskii have exhaustively treated the case where H has a homogeneous spectrum and is of Szegö class. In [2] Egorova and the authors have established direct and inverse scattering theory for Jacobi operators which are short range perturbations of quasi-periodic finite-gap operators. For further information and references we refer to these articles and [12].

In the case of constant background it is well-known that the transmission coefficient is the perturbation determinant in the sense of Krein [8], see e.g., [11] or [12]. The purpose of the present paper is to establish this result for the case of quasi-periodic finite-gap background, thereby establishing the connection with Krein’s spectral shift theory. For related results see also [7], [10].

Moreover, scattering theory for Jacobi operators is not only interesting in its own right, it also constitutes the main ingredient of the inverse scattering transform for the Toda hierarchy (see, e.g., [5], [4], [12], or [13]). Since the transmission coefficient is invariant when our Jacobi operator evolves in time with respect to some equation of the Toda hierarchy, the corresponding trace formulas provide the conserved quantities for the Toda hierarchy in this setting.

1991 Mathematics Subject Classification. Primary 47B36, 37K15; Secondary 81U40, 39A11.

Key words and phrases. Scattering, Toda hierarchy, Trace formulas.

Work supported by the Austrian Science Fund (FWF) under Grant No. P17762.

2. Notation

We assume that the reader is familiar with quasi-periodic Jacobi operators. Hence we only briefly recall some notation and refer to [2] and [12] for further information.

Let

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

be a quasi-periodic Jacobi operator in ℓ^{2}(ℤ) associated with the Riemann surface of the function

(2.2) R_{2g+2}^{1/2}(z),  R_{2g+2}(z) = ∏_{j=0}^{2g+1}(z − E_{j}),  E_{0} < E_{1} < ⋯ < E_{2g+1},

g ∈ ℕ. The spectrum of H_{q} is purely absolutely continuous and consists of g + 1 bands

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

For every z ∈ ℂ the Baker-Akhiezer functions ψ_{q,±}(z, n) are two (weak) solutions of H_{q}ψ = zψ, which are linearly independent away from the band-edges {E_{j}}_{j=0}^{2g+1}, since their Wronskian is given by

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

Here μ_{j} are the Dirichlet eigenvalues at base point n_{0} = 0. We recall that ψ_{q,±}(z, n) have the form

ψ_{q,±}(z, n) = θ_{q,±}(z, n)w(z)^{±n},

where θ_{q,±}(z, n) is quasi-periodic with respect to n and w(z) is the quasi-momentum. In particular, |w(z)| < 1 for z ∈ ℂ\σ(H_{q}) and |w(z)| = 1 for z ∈ σ(H_{q}).

3. Asymptotics of Jost solutions

After we have these preparations out of our way, we come to the study of shortrange perturbations H of H_{q} associated with sequences a, b satisfying a(n) → a_{q}(n) and b(n) → b_{q}(n) as |n| → ∞. More precisely, we will make the following assumption throughout this paper:

Let H be a perturbation of H_{q} such that

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

that is, H − H_{q} is trace class.

We first establish existence of Jost solutions, that is, solutions of the perturbed operator which asymptotically look like the Baker-Akhiezer solutions.

Theorem 3.1. Assume (3.1). For every z ∈ ℂ\{E_{j}}_{j=0}^{2g+1} there exist (weak) solutions ψ_{±}(z, .) of Hψ = zψ satisfying

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

where ψ_{q,±}(z, .) are the Baker-Akhiezer functions. Moreover, ψ_{±}(z, .) are continuous (resp. holomorphic) with respect to z whenever ψ_{q,±}(z, .) are, and have the following asymptotic behavior (3.3)

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

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

Note that since a_{q}(n) are bounded away from zero, A_{±}(n) are well-defined. Here

the star indicates that \begin{matrix}n \\ ∑ & ∗ \\ j=1 \end{matrix} = − ∑_{j=n+1}^{0} for n < 0 and similarly for the product.

Proof. The proof can be done as in the periodic case (see e.g., [2], [6], [9] or [12], Section 7.5). There a stronger decay assumption (i.e., first moments summable) is made, which is however only needed at the band edges {E_{j}}_{j=0}^{2g+1}.

For later use we note the following immediate consequence

Corollary 3.2. Under the assumptions of the previous theorem we have

lim_{n→±∞} w(z)^{∓n} (ψ′_{±}(z, n) ∓ n\frac{w′(z)}{w(z)}ψ_{±}(z, n) − ψ′_{q,±}(z, n) ± n\frac{w′(z)}{w(z)}ψ_{q,±}(z, n)) = 0,

where the prime denotes differentiation with respect to z.

Proof. Just differentiate (3.2) with respect to z, which is permissible by uniform convergence on compact subsets of ℂ\{E_{j}}_{j=0}^{2g+1}.

We remark that if we require our perturbation to satisfy the usual short range assumption as in [2] (i.e., first moments summable), then we even have w(z)^{∓n}(ψ′_{±}(z, n)− ψ′_{q,±}(z, n)) → 0.

From Theorem 3.1 we obtain a complete characterization of the spectrum of H.

Theorem 3.3. Assume (3.1). Then we have σ_{ess}(H) = σ(H_{q}), the point spectrum of H is confined to \overline{ℝ\σ(H_{q})}. Furthermore, the essential spectrum of H is purely absolutely continuous except for possible eigenvalues at the band edges.

Proof. This is an immediate consequence of the fact that H − H_{q} is trace class and boundedness of the Jost solutions inside the essential spectrum.

Our next result concerns the asymptotics of the Jost solutions at the other side.

Lemma 3.4. Assume (3.1). Then the Jost solutions ψ_{±}(z, .), z ∈ ℂ\σ(H), satisfy

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

where

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

Proof. Since H − H_{q} is trace class, we have for the difference of the Green’s functions

lim_{n→±∞} G(z, n, n) − G_{q}(z, n, n) = lim_{n→±∞}⟨δ_{n}, ((H − z)^{−1} − (H_{q} − z)^{−1})δ_{n}⟩ = 0

and using

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

we obtain

lim_{n→−∞} ψ_{q,−}(z, n)(ψ_{+}(z, n) − α(z)ψ_{q,+}(z, n)) = 0,

which is the claimed result.

Note that α(z) is just the inverse of the transmission coefficient (see, e.g., [2] or [12], Section 7.5). It is holomorphic in ℂ\σ(H_{q}) with simple zeros at the discrete eigenvalues of H and has the following asymptotic behavior

(3.7) α(z) = \frac{1}{A}(1 + \frac{B}{z} + O(z^{−2})), A = A_{−}(0)A_{+}(0), B = B_{−}(1) + B_{+}(0),

with A_{±}(n), B_{±}(n) from (3.4).

4. Connections with Krein’s spectral shift theory and trace formulas

To establish the connection with Krein’s spectral shift theory we next show:

Lemma 4.1. We have

(4.1)  \frac{d}{dz} α(z) = −α(z) ∑_{n∈ℤ} (G(z, n, n) − G_{q}(z, n, n)),  z ∈ ℂ\σ(H),

where G(z, m, n) and G_{q}(z, m, n) are the Green’s functions of H and H_{q}, respectively.

Trace Formulas for Jacobi Operators 5

Proof. Green’s formula ([12], eq. (2.29)) implies

(4.2)  W_{n}(ψ_{+}(z), ψ′_{−}(z)) − W_{m−1}(ψ_{+}(z), ψ′_{−}(z)) = ∑_{j=m}^{n} ψ_{+}(z, j)ψ_{−}(z, j),

hence the derivative of the Wronskian can be written as

\frac{d}{dz} W (ψ_{−}(z), ψ_{+}(z)) = W_{n}(ψ′_{−}(z), ψ_{+}(z)) + W_{n}(ψ_{−}(z), ψ′_{+}(z))

= W_{m}(ψ′_{−}(z), ψ_{+}(z)) + W_{n}(ψ_{−}(z), ψ′_{+}(z)) − ∑_{j=m+1}^{n} ψ_{+}(z, j)ψ_{−}(z, j).

Using Corollary 3.2 and Lemma 3.4 we have

W_{m}(ψ′_{−}(z), ψ_{+}(z)) = W_{m}(ψ′_{−} + m\frac{w′}{w} ψ_{−}, ψ_{+}) −

\frac{w′}{w} (m W (ψ_{−}, ψ_{+}) − a(m)ψ_{−}(m + 1)ψ_{+}(m))

→ αW_{q,m}(ψ′_{q,−} + m\frac{w′}{w} ψ_{q,−}, ψ_{q,+}) −

α\frac{w′}{w} (m W_{q}(ψ_{q,−}, ψ_{q,+}) − a_{q}(m)ψ_{q,−}(m + 1)ψ_{q,+}(m))

= α(z)W_{m}(ψ′_{q,−}(z), ψ_{q,+}(z))

as m → −∞. Similarly we obtain

W_{n}(ψ_{−}(z), ψ′_{+}(z)) → α(z)W_{n}(ψ_{q,−}(z), ψ′_{q,+}(z))

as n → ∞ and again using (4.2) we have

W_{m}(ψ′_{q,−}(z), ψ_{q,+}(z)) = W_{n}(ψ′_{q,−}(z), ψ_{q,+}(z)) + ∑_{j=m+1}^{n} ψ_{q,+}(z, j)ψ_{q,−}(z, j).

Collecting terms we arrive at

W′(ψ_{−}(z), ψ_{+}(z)) = − ∑_{j∈ℤ} (ψ_{+}(z, j)ψ_{−}(z, j) − α(z)ψ_{q,+}(z, j)ψ_{q,−}(z, j))
+ α(z)W′_{q}(ψ_{q,−}(z)ψ_{q,+}(z)).

Now we compute

\frac{d}{dz} α(z) = \frac{d}{dz} (\frac{W}{W_{q}} ) = (\frac{1}{W_{q}} )′W + \frac{1}{W_{q}} W′

= −\frac{W′_{q}}{W_{q}^{2}} W + \frac{1}{W_{q}} ( − ∑_{j∈ℤ} (ψ_{+}ψ_{−} − αψ_{q,+}ψ_{q,−}) + αW′_{q})

= −\frac{1}{W_{q}} ∑_{j∈ℤ} (ψ_{+}ψ_{−} − αψ_{q,+}ψ_{q,−}),

which finishes the proof.

As an immediate consequence, we can identify α(z) as Krein’s perturbation determinant ([8]) of the pair H, H_{q}.

Theorem 4.2. The function Aα(z) is Krein’s perturbation determinant:

(4.3) α(z) = \frac{1}{A} det (1l + (H(t) − H_{q}(t))(H_{q}(t) − z)^{−1}),  A = ∏_{j∈ℤ} \frac{a(j)}{a_{q}(j)}.

By [8], Theorem 1, α(z) has the following representation

(4.4)  α(z) = \frac{1}{A} exp ( ∫_{ℝ} \frac{ξ_{α}(λ)dλ}{λ − z}),

where

(4.5)  ξ_{α}(λ) = \frac{1}{π} lim_{ϵ↓0} arg α(λ + iϵ)

is the spectral shift function.

Hence

(4.6)  τ_{j} = tr(H^{j} − (H_{q})^{j}) = j ∫_{ℝ} λ^{j−1}ξ_{α}(λ)dλ,

where τ_{j}/j are the expansion coefficients of ln α(z) around z = ∞:

ln α(z) = − ln A − ∑_{j=1}^{∞} \frac{τ_{j}}{j z^{j}} .

They are related to the expansion α_{j} coefficients of

α(z) = \frac{1}{A} ∑_{j=0}^{∞} \frac{α_{j}}{z^{j}} , α_{0} = 1,

via

(4.7)  τ_{1} = −α_{1},  τ_{j} = −jα_{j} − ∑_{k=1}^{j−1} α_{j−k}τ_{k}.

5. Conserved quantities of the Toda hierarchy

Finally we turn to solutions of the Toda hierarchy TL_{r} (see, e.g., [1], [4], [12], or [13]). Let (a_{q}(t), b_{q}(t)) be a quasi-periodic finite-gap solution of some equation in the Toda hierarchy, TL_{r}(a_{q}(t), b_{q}(t)) = 0, and let (a(t), b(t)) be another solution, TL_{r}(a(t), b(t)) = 0, such that (3.1) holds for one (hence any) t.

Since the transmission coefficient T (z, t) = T (z, 0) ≡ T (z) is conserved (see [3] – formally this follows from unitary invariance of the determinant), so is α(z) = T (z)^{−1}.

Trace Formulas for Jacobi Operators 7

Theorem 5.1. The quantities

(5.1)  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})

⋮

are conserved quantities for the Toda hierarchy.

Acknowledgments

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

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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/