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.