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

In this paper we investigate the one-dimensional Schrodinger operator L(q) with complex-valued periodic potential q when q\inL_{1}[0,1] and q_{n}=0 for n=0,-1,-2,..., where q_{n} are the Fourier coefficients of q with respect to the system {e^{i2{\pi}nx}}. We prove that the Bloch eigenvalues are (2{\pi}n+t)^{2} for n\inZ, t\inC and find explicit formulas for the Bloch functions. Then we consider the inverse problem for this operator.

We investigate the multidimensional Schrodinger operator L(q) with complex-valued periodic, with respect to a lattice, potential q when the Fourier coefficients of q with respect to the orthogonal system {exp(i(a,x))}, where a changes in the dual lattice, vanish if a belong to a half-space We prove that the Bloch eigenvalues of L(q) and of the free operator L(0) are the same and find explicit formulas for the Bloch functions. It implies that the Fermi surfaces of L(q) and L(0...

We use B\'{e}zout's theorem and Bernstein-Khovanskii-Kushnirenko theorem to analyze the level sets of the extrema of the spectral band functions of discrete periodic Schr\"odinger operators on $\mathbb{Z}^2$. These approaches improve upon previous results of Liu and Filonov-Kachkovskiy.

According to its Lax pair formulation, the nonlinear Schr\"odinger (NLS) equation can be expressed as the compatibility condition of two linear ordinary differential equations with an analytic dependence on a complex parameter. The first of these equations---often referred to as the \emph{$x$-part} of the Lax pair---can be rewritten as an eigenvalue problem for a Zakharov-Shabat operator. The spectral analysis of this operator is crucial for the solution of the initial value ...

We consider Schr\"odinger operators with periodic potentials in the positive quadrant for dim $>1$ with Dirichlet boundary condition. We show that for any integer $N$ and any interval $I$ there exists a periodic potential such that the Schr\"odinger operator has $N$ eigenvalues counted with the multiplicity on this interval and there is no other spectrum on the interval. Furthermore, to the right and to the left of it there is a essential spectrum. Moreover, we prove simila...

We discuss applications of the M. G. Kre\u{\i}n theory of the spectral shift function to the multi-dimensional Schr\"odinger operator as well as specific properties of this function, for example, its high-energy asymptotics. Trace identities for the Schr\"odinger operator are derived.

In this article we obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and quasiperiodic boundary conditions. Then using these asymptotic formulas, we find conditions on the coefficients for which the number of gaps in the spectrum of the self-adjoint differential operator with the periodic matrix coefficients is finite.

The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which influence significantly most properties of such operators. The approaches described are applicable not only to the standard model example of Schr\"odinger operator with periodic electric potential $-\Delta+V(x)$, but to a wide variety of el...

We consider normalized Laplacians and their perturbations by periodic potentials (Schr\"odinger operators) on periodic discrete graphs. The spectrum of the operators consists of an absolutely continuous part (a union of a finite number of non-degenerate bands) and a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We obtain estimates of the Lebesgue measure of the spectrum in terms of geometric parameters of the graphs and show that they become identit...

In this article we discuss a procedure to solve the one dimensional (1D) Schroedinger Equation for a periodic potential, which may be well suited to teach band structure theory. The procedure is conceptually very simple, so that it may be used to teach band theory at the undergraduate level; at the same time the point of view is practical, so that the students may experiment computing band gaps, and other features of band structure. Another advantage of the procedure lies in ...