### A study of resolvent set for a class of band operators with matrix elements

For operators generated by a certain class of infinite three-diagonal matrices with matrix elements we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying second order finite-difference equations. This enables us to describe some asymptotic behavior of the corresponding systems of vector orthogonal polynomials on the resolvent set. We also find that the operators generated by infinite Jacobi matrices have the largest resolvent set in this class.

### A toroidal compactification of the Fermi surface for the discrete Schrödinger operator.

### Absence of point spectrum for a class of discrete Schrödinger operators with quasiperiodic potential.

### Absolutely continuous spectrum and scattering in the surface Maryland model

We study the discrete Schrödinger operator $H$ in ${\mathbf{Z}}^{d}$ with the surface quasi periodic potential $V\left(x\right)=g\delta \left({x}_{1}\right)tan\pi (\alpha \xb7{x}_{2}+\omega )$, where $x=({x}_{1},{x}_{2}),\phantom{\rule{4pt}{0ex}}{x}_{1}\in {\mathbf{Z}}^{{d}_{1}},\phantom{\rule{4pt}{0ex}}{x}_{2}\in {\mathbf{Z}}^{{d}_{2}},\phantom{\rule{4pt}{0ex}}\alpha \in {\mathbf{R}}^{{d}_{2}},\phantom{\rule{4pt}{0ex}}\omega \in [0,1)$. We first discuss a proof of the pure absolute continuity of the spectrum of $H$ on the interval $[-d,d]$ (the spectrum of the discrete laplacian) in the case where the components of $\alpha $ are rationally independent. Then we show that in this case the generalized eigenfunctions have the form of the “volume” waves, i.e. of the sum of the incident plane wave and reflected from the hyper-plane ${\mathbf{Z}}^{{d}_{1}}$ waves, the form...

### Almost localization and almost reducibility

### An index theorem for systems of difference operators on a half space

### An inverse problem for the discrete periodic Schrödinger operator.

### Cheeger inequalities for unbounded graph Laplacians

We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.

### Criterion of $p$-criticality for one term $2n$-order difference operators

We investigate the criticality of the one term $2n$-order difference operators $l{\left(y\right)}_{k}={\Delta}^{n}\left({r}_{k}{\Delta}^{n}{y}_{k}\right)$. We explicitly determine the recessive and the dominant system of solutions of the equation $l{\left(y\right)}_{k}=0$. Using their structure we prove a criticality criterion.

### Discrete spectra criteria for singular difference operators

We investigate oscillation and spectral properties (sufficient conditions for discreteness and boundedness below of the spectrum) of difference operators B(y)n+k = (-1)nwk n (pk n yk).

### Eigenvalue curves of asymmetric tridiagonal random matrices.

### Extensions, dilations and functional models of infinite Jacobi matrix

A space of boundary values is constructed for the minimal symmetric operator generated by an infinite Jacobi matrix in the limit-circle case. A description of all maximal dissipative, accretive and selfadjoint extensions of such a symmetric operator is given in terms of boundary conditions at infinity. We construct a selfadjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation....

### Fonctions de mailles et théorie elliptique des opérateurs aux différences finies (suite)

### Fonctions de mailles et théorie elliptique des opérateurs aux différences finies

### Fredholm determinants and the Evans function for difference equations

We develop a difference equations analogue of recent results by F. Gesztesy, K. A. Makarov, and the second author relating the Evans function and Fredholm determinants of operators with semi-separable kernels.

### Fundamental operator-functions of degenerate differential and difference-differential operators in Banach spaces which have a Noether operator in the principal part.

### Internal Lifshitz tails for discrete Schrödinger operators.

### L//p-оценки решений одной разностной краевой задачи.

### Metal-insulator transition for the almost Mathieu operator.