The existence of eigenvalues of Schrödinger operator on a lattice in the gap of the essential spectrum

We consider a three-particle discrete Schrödinger operator Hμγ (K), K 2 T3 associated to a system of three particles (two fermions and one different particle) interacting through zero range pairwise potential μ > 0 on the three-dimensional lattice Z 3. It is proved that the operator Hμγ (K), ||K|| < δ, for γ > γ0 has at least two eigenvalues in the gap of the essential spectrum for sufficiently large μ > 0.

Ambarzumian-type Problems for Discrete Schrödinger Operators

We discuss the problem of unique determination of the finite free discrete Schrödinger operator from its spectrum, also known as the Ambarzumian problem, with various boundary conditions, namely any real constant boundary condition at zero and Floquet boundary conditions of any angle. Then we prove the following Ambarzumian-type mixed inverse spectral problem: diagonal entries except the first and second ones and a set of two consecutive eigenvalues uniquely determine the finite free discrete Schrödinger operator.

A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry

The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the 2M-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new polynomial identity relating the eigenvalues of such matrices with their matrix entries is obtained. In the limit M→∞ this identity induces some requirements, which should satisfy the scattering data of the resulting infinite-dimensional Jacobi operator in the half-line, of which super- and subdiagonal matrix elements are equal to -1. We obtain such requirements in the simplest case of the discrete Schrödinger operator acting in l2(N), which does not have bound and semibound states and whose potential has a compact support.

Analytic Scattering Theory for Jacobi Operators and Bernstein–Szegö Asymptotics of Orthogonal Polynomials

We study semi-infinite Jacobi matrices [Formula: see text] corresponding to trace class perturbations [Formula: see text] of the “free” discrete Schrödinger operator [Formula: see text]. Our goal is to construct various spectral quantities of the operator [Formula: see text], such as the weight function, eigenfunctions of its continuous spectrum, the wave operators for the pair [Formula: see text], [Formula: see text], the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials [Formula: see text] associated to the Jacobi matrix [Formula: see text] as [Formula: see text]. In particular, we consider the case of [Formula: see text] inside the spectrum [Formula: see text] of [Formula: see text] when this asymptotic has an oscillating character of the Bernstein–Szegö type and the case of [Formula: see text] at the end points [Formula: see text].