Trace Formulas for Jacobi Operators in Connection with Scattering Theory for Quasi-Periodic Background (original) (raw)
Scattering Theory for Jacobi Operators with Quasi-Periodic Background
Communications in Mathematical Physics, 2006
We develop direct and inverse scattering theory for Jacobi operators which are short range perturbations of quasi-periodic finite-gap operators. We show existence of transformation operators, investigate their properties, derive the corresponding Gel'fand-Levitan-Marchenko equation, and find minimal scattering data which determine the perturbed operator uniquely.
Scattering theory for Jacobi operators with a steplike quasi-periodic background
Inverse Problems, 2007
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.
Trace formulas and inverse spectral theory for finite Jacobi operators
Based on high energy expansions and Herglotz properties of Green and Weyl m-functions we develop a self-contained theory of trace formulas for Jacobi operators. In addition, we consider connections with inverse spectral theory, in particular uniqueness results. As an application we work out a new approach to the inverse spectral problem of a class of reflectionless operators producing explicit formulas for the coefficients in terms of minimal spectral data. Finally, trace formulas are applied to scattering theory with periodic backgrounds.
Soliton solutions of the Toda hierarchy on quasi-periodic backgrounds revisited
Mathematische Nachrichten, 2009
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.
Periodic Jacobi operators with finitely supported perturbations. arXiv
2016
We consider a periodic Jacobi operator J with finitely supported perturbations on the half-lattice. We describe all eigenvalues and resonances of J and give their properties. We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the Jost functions is one-to-one and onto, we show how the Jost functions can be reconstructed from all eigenvalues, resonances and from the set of zeros of S(λ) − 1, where S(λ) is the scattering matrix.
Resonances for periodic Jacobi operators with finitely supported perturbations
Journal of Mathematical Analysis and Applications, 2012
We describe the spectral properties of the Jacobi operator (Hy) n = a n−1 y n−1 +a n y n+1 + b n y n , n ∈ Z, with a n = a 0 n + u n , b n = b 0 n + v n , where sequences a 0 n > 0, b 0 n ∈ R are periodic with period q, and sequences u n , v n have compact support. In the case u n ≡ 0 we obtain the asymptotics of the spectrum in the limit of small perturbations v n .
Periodic Jacobi operator with finitely supported perturbations: the inverse resonance problem
2011
We consider a periodic Jacobi operator H with finitely supported perturbations on Z. We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the scattering data: the inverse of the transmission coefficient and the Jost function on the right half-axis, is one-to-one and onto. We consider the problem of reconstruction of the scattering data from all eigenvalues, resonances and the set of zeros of R_-(λ)+1, where R_- is the reflection coefficient.
On the spectrum of Jacobi operators with quasiperiodic algebro-geometric coefficients
2005
We characterize the spectrum of one-dimensional Jacobi operators H = aS + + a − S − + b in l 2 (Z) with quasi-periodic complex-valued algebrogeometric coefficients (which satisfy one (and hence infinitely many) equation(s) of the stationary Toda hierarchy) associated with nonsingular hyperelliptic curves. The spectrum of H coincides with the conditional stability set of H and can explicitly be described in terms of the mean value of the Green's function of H.
Periodic Jacobi operator with finitely supported perturbation on the half-lattice
Inverse Problems, 2011
We consider a periodic Jacobi operator J with finitely supported perturbations on the half-lattice. We describe all eigenvalues and resonances of J and give their properties. We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the Jost functions is one-to-one and onto, we show how the Jost functions can be reconstructed from all eigenvalues, resonances and from the set of zeros of S(λ) − 1, where S(λ) is the scattering matrix.
A local trace formula for resonances of perturbed periodic Schrödinger operators
Journal of Functional Analysis, 2003
We give a local trace formula for the pair ðP 1 ðhÞ ¼ P 0 þ W ðhyÞ; P 0 Þ; where P 0 is a periodic Schro¨dinger operator, W is a decreasing perturbation and h is a small positive parameter. We apply this result to establish the existence of Bh Àn resonances near some energy l of sðP 0 Þ:
Spectral results for perturbed periodic Jacobi matrices using the discrete Levinson technique
Studia Mathematica, 2018
For an arbitrary Hermitian period-T Jacobi operator, we assume a perturbation by a Wigner-von Neumann type potential to devise subordinate solutions to the formal spectral equation for a (possibly infinite) real set, S, of the spectral parameter. We employ discrete Levinson type techniques to achieve this, which allow the analysis of the asymptotic behaviour of the solution. This enables us to construct infinitely many spectral singularities on the absolutely continuous spectrum of the periodic Jacobi operator, which are stable with respect to an l 1-perturbation. An analogue of the quantisation conditions from the continuous case appears, relating the frequency of the oscillation of the potential to the quasi-momentum associated with the purely periodic operator.
Journal of Differential Equations, 2003
Dene the periodic weighted operator T y= 2 ( 2 y 0 ) 0 in L 2 (R; ( x ) 2 dx). Suppose a function 2 W 2 1 (R=Z) is 1-periodic real positive and let q = 0 = 2 L 2 (0; 1). The spectrum of T consists of intervals n = [ n 1 ; + n ] and + 0 = 0, separated by gaps n = ( n ; + n ) ; n>1 ;with the lengths j n j. Let h n be a height of the corresponding slit in the quasimomentum domain and let g n ; n>1 ;be the gap with the length jg n j of the operator p T > 0. Introduce the sequences = fj n jg; h=f h n g ; g=fjg n jg, and the norms kfk 2 m = P n>1 (2n) 2m f 2 n ; m2Z .The following results are obtained: i) the quasimomentum k for the weighted operator T is constructed and the needed properties of k have been studied, ii) double-side estimates of kk 1 ; khk; kgk in terms of kqk 2 = R 1 0 q(x) 2 dx, iii) the asymptotics of gap length j n j as n ! 1 , iv) let the function q 2 C 1 (0; 1) and q 0 (x) < 0; x2(0; 1). Then each gap of T is nondegenerate. The proofs are based on the analysis of the quasimomentum as the conformal mapping, the embedding theorems and the identities between the quasimomentum and the potential. In order to prove these results the asymptotics of the fundamental solutions and the Lyapunov function are found at high energy .
Eigenvalues for Perturbed Periodic Jacobi Matrices by the Wigner-von Neumann Approach
Integral Equations and Operator Theory, 2016
The Wigner-von Neumann method, which has previously been used for perturbing continuous Schrödinger operators, is here applied to their discrete counterparts. In particular, we consider perturbations of arbitrary T-periodic Jacobi matrices. The asymptotic behaviour of the subordinate solutions is investigated, as too are their initial components, together giving a general technique for embedding eigenvalues, λ, into the operator's absolutely continuous spectrum. Introducing a new rational function, C(λ; T), related to the periodic Jacobi matrices, we describe the elements of the a.c. spectrum for which this construction does not work (zeros of C(λ; T)); in particular showing that there are only finitely many of them.
Marchenko-Ostrovski mappings for periodic Jacobi matrices
Russian Journal of Mathematical Physics, 2007
We consider the 1D periodic Jacobi matrices. The spectrum of this operator is purely absolutely continuous and consists of intervals separated by gaps. We solve the inverse problem (including characterization) in terms of vertical slits on the quasimomentum domain. Furthermore, we obtain a priori two-sided estimates for vertical slits in terms of Jacoby matrices.
Inverse spectral theory for Jacobi matrices and their almost periodicity
1994
In this paper we consider the inverse problem for bounded Jacobi matrices with nonempty absolutely continuous spectrum and as an application show the almost periodicity of some random Jacobi matrices. We do the inversion in two different ways. In the general case we use a direct method of reconstructing the Green functions. In the special case where we show the almost periodicity, we use an alternative method using the trace formula for points in the orbit of the matrices under translations. This method of reconstruction involves analyzing the Abel-Jacobi map and solving of the Jacobi inversion problem associated with an infinite genus Riemann surface constructed from the spectrum. Contents 1 Introduction 2 1.1 Ideas, strategies and limitations : : : : : : : : : : : : : : : : : 4 2 Inverse Spectral Theory 8 3 Interpolation theorem 18 4 Analysis on a Riemann surface 32 4.1 The Riemann Surface : : : : : : : : : : : : : : : : : : : : : : : 32 4.2 The Abel-Jacobi map : : : : : : : : :...
Embedded eigenvalues for perturbed periodic Jacobi operators using a geometric approach
Journal of Difference Equations and Applications, 2018
We consider the problem of embedding eigenvalues into the essential spectrum of periodic Jacobi operators, using an oscillating, decreasing potential. To do this we employ a geometric method, previously used to embed eigenvalues into the essential spectrum of the discrete Schrödinger operator. For periodic Jacobi operators we relax the rational dependence conditions on the values of the quasi-momenta from this previous work. We then explore conditions that permit not just the existence of infinitely many subordinate solutions to the formal spectral equation but also the embedding of infinitely many eigenvalues.