SEMICLASSICAL ASYMPTOTICS OF EIGENVALUES FOR NON-SELFADJOINT OPERATORS AND QUANTIZATION CONDITIONS ON RIEMANN SURFACES

Anna I Esina,Andrei I Shafarevich

doi:10.14311/ap.2014.54.0101

Abstract

This paper reports a study of the semiclassical asymptotic behavior of the eigenvalues of some nonself-adjoint operators that are important for applications. These operators are the Schrödinger operator with complex periodic potential and the operator of induction. It turns out that the asymptotics of the spectrum can be calculated using the quantization conditions. These can be represented as the condition that the integrals of a holomorphic form over the cycles on the corresponding complex Lagrangian manifold, which is a Riemann surface of constant energy, are integers. In contrast to the real case (the Bohr–Sommerfeld–Maslov formulas), in order to calculate a chosen spectral series, it is sufficient to assume that the integral over only one of the cycles takes integer values, and different cycles determine different parts of the spectrum.

Highlights

IntroductionOne of the main problems of the semiclassical theory (see, for example, [1]) is the description of the asymptotic behavior of the spectrum of operators of the form
One of the main problems of the semiclassical theory is the description of the asymptotic behavior of the spectrum of operators of the form H = (x, −ıh ∂ ∂x ), h →In this case, the problem can naturally be divided into two subproblems, namely:(1.) to solve the spectral equation approximately, i.e., to find numbers λ and functions ψ, satisfying the following equation for some N > 1: Hψ = λψ + O(hN ); (1)
The Riemann surface Λ is given by the equation p2 + ina(z) = λ (n is an integer constant entering the separation of variables), and the spectral graph is defined from equations (9), in which V = na

Summary

Introduction

One of the main problems of the semiclassical theory (see, for example, [1]) is the description of the asymptotic behavior of the spectrum of operators of the form. The function ψ mentioned in the theorem can be described in a computable way, namely, it is of the form K(1), where K stands for the Maslov canonical operator on the Liouville torus Λ. We consider two classes of nonselfadjoint operators, namely, the one-dimensional Schrödinger operator with complex potential and the operator of magnetic induction on a two-dimensional symmetric surface The spectrum of these operators, in the semiclassical limit, is concentrated in the O(h2)neighborhood of some curves in the complex plane E; these curves form the so-called spectral graph. In contrast to the self-adjoint case, in order to evaluate the eigenvalues, it is required to satisfy the corresponding condition on only one cycle, and it turns out that different cycles determine different parts of the spectrum (and different edges of the spectral graph)

Schrödinger equation with a complex potential

Equation of magnetic induction

Conclusions