On the Constructively Determination the Spectral Invariants of the Periodic Multidimensional Schrodinger Operator

We study how the spectral properties of ergodic Schr\"odinger operators are reflected in the asymptotic properties of its periodic approximation as the period tends to infinity. The first property we address is the asymptotics of the bandwidths on the logarithmic scale, which quantifies the sensitivity of the finite volume restriction to the boundary conditions. We show that the bandwidths can always be bounded from below in terms of the Lyapunov exponent. Under an additional...

We survey Barry Simon's principal contributions to the field of inverse spectral theory in connection with one-dimensional Schrodinger and Jacobi operators.

In the present paper our aim is to explore some spectral properties of the family two-particle discrete Schr\"odinger operators $h^{\mathrm{d}}(k)=h^{\mathrm{d}}_0(k)+ \mathbf{v},$ $k\in \T^\mathrm{d},$ on the $\mathrm{d}$ dimensional lattice $\Z^{\mathrm{d}},$ $\mathrm{d}\geq 1,$ $k$ being the two-particle quasi-momentum. Under some condition in the case $k\in \T^{\mathrm{d}}\setminus (-\pi,\pi)^{\mathrm{d}},$ we establish necessary and sufficient conditions for existence of...

The determination of the spectrum of a Schr\"odinger operator is a fundamental problem in mathematical quantum mechanics. We discuss a series of results showing that Schr\"odinger operators can exhibit spectra that are remarkably thin in the sense of Lebesgue measure and fractal dimensions. We begin with a brief discussion of results in the periodic theory, and then move to a discussion of aperiodic models with thin spectra.

In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schr\"odinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an introductory part explaining basic spectral concepts and fundamental results, we present the general theory of such operators, and then provide an overview of known results for specific classes of potentials. Here we focus primarily on the...

We survey results concerning the spectral properties of limit-periodic operators. The main focus is on discrete one-dimensional Schr\"odinger operators, but other classes of operators, such as Jacobi and CMV matrices, continuum Schr\"odinger operators and multi-dimensional Schr\"odinger operators, are discussed as well. We explain that each basic spectral type occurs, and it does so for a dense set of limit-periodic potentials. The spectrum has a strong tendency to be a Can...

We consider discrete Schr\"odinger operators with periodic potentials on periodic graphs perturbed by guided non-positive potentials, which are periodic in some directions and finitely supported in other ones. The spectrum of the unperturbed operator is a union of a finite number of non-degenerate bands and eigenvalues of infinite multiplicity. We show that the spectrum of the perturbed operator consists of the unperturbed one plus the additional guided spectrum, which is a u...

We study the direct and inverse spectral problems for semiclassical operators of the form $S = S_0 +\h^2V$, where $S_0 = \frac 12 \Bigl(-\h^2\Delta_{\bbR^n} + |x|^2\Bigr)$ is the harmonic oscillator and $V:\bbR^n\to\bbR$ is a tempered smooth function. We show that the spectrum of $S$ forms eigenvalue clusters as $\h$ tends to zero, and compute the first two associated "band invariants". We derive several inverse spectral results for $V$, under various assumptions. In particul...

We consider a Laplacian on periodic discrete graphs. Its spectrum consists of a finite number of bands. In a class of periodic 1-forms, i.e., functions defined on edges of the periodic graph, we introduce a subclass of minimal forms with a minimal number $\cI$ of edges in their supports on the period. We obtain a specific decomposition of the Laplacian into a direct integral in terms of minimal forms, where fiber Laplacians (matrices) have the minimal number $2\cI$ of coeffic...

We consider the Schr\"odinger operator on the real line with a $N\ts N$ matrix valued periodic potential, N>1. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define the Lyapunov function, which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. The Lyapunov function has (real or complex) branch points, which...