The approximate spectral projection method

Abstract

In this paper we develop a general method for investigating the spectral asymptotics for various differential and pseudo-differential operators and their boundary value problems, and consider many of the problems posed when this method is applied to mathematical physics and mechanics. Among these problems are the Schr6dinger operator with growing, decreasing and degenerating potential, the Dirac operator with decreasing potential, the 'quasi-classical' spectral asymptotics for Schr6dinger and Dirac operators, the linearized Navier-Stokes equation, the Maxwell system, the system of reactor kinetics, the eigenfrequency problems of shell theory, and so on. The method allows us to compute the principal term of the spectral asymptotics (and, in the case of Douglis-Nirenberg elliptic operators, also their following terms) with the remainder estimate close to that for the sharp remainder. AMS (MOS) subject classification (1980). 35P30.Key words. Spectral asymptotics, remainder estimate, differential and pseudo-differential operators, boundary value problems. 0. Introduction 0.1. Eigenvalue problems for self-adjoint differential operators arise in many fields of mathematical physics and mechanics. Thus, the eigenvalues of a Schr6dinger operator are the energy levels of a quantum particle, and the eigenvalues of the Maxwell system, of the linearized Navier-Stokes equation, of the shell-theory operator and so on, are the squares of the eigenfrequencies of the corresponding bodies.It is clear that the computation of the eigenvalues of self-adjoint operators is a problem of great importance, but it is usually impossible to compute them exactly, except in some simple cases. Hence, approximate methods of computation are of great interest.One of these methods is the investigation of the asymptotics of the series of eigenvalues An of the operator A, as n--~ ~. If A is an operator with a discrete spectrum, then usually it is more convenient to study the asymptotics of the distribution functions of the series A~ of the positive and negative eigenvalues: N±(t, A) = card{i I + )t~: < t}, as t---~ ~.If the series {An} ~ [a, b), )tn ~ b, and on this interval there are points of a discrete spectrum only, then we can study the asymptotics of the function N((a, b -T), A) = card{/I a < Ai < b -z}, as T---~ +0.

Asymptotic formulae with remainder estimates for eigenvalue branches of the Schrödinger operator $H - \lambda W$ in a gap of $H$

“…(See also [14], where asymptotic formulae for pdo of negative order were derived from Theorem 6.2.) By gathering (3.14)-(3.18) we obtain…”

Eigenvalue Asymptotics of Schrödinger Operators with Only Discrete Spectrum

“…For the results of the form of (1.1) we refer to Edmunds and Evans [2], Feigin [4], Fleckinger [5], Fleckinger and Lapidus [6, Section 5], Levendorskii [9], Reed and Simon [10,Theorem XIII. 81], Rozenbljum [12], Tamura [16], Titchmarsh [17, Chapter XVII] and de Wet and Mandl [18].…”

“…A nonclassical potential is the potential V such that F>0 and the set {xeR n : V(x) = 0} is an unbounded set. Several results on these nonclassical potentials are known ( [8], [9,Section 10] [11], [13], [14], [15]) and those are only on the potentials whose zero sets are cones in R". Our method is a modification of that of Tachizawa [15] and we can apply it to some nonclassical potentials whose zero sets are not cones.…”

Spectral asymptotics with a remainder estimate for Schrödinger operators with slowly growing potentials

“…There are few exceptions (see e.g. [1,4,7]) where q is allowed to grow very slowly, e.g. q(x) = In .…”

“…. In |x| outside some compact, but the conditions in [1] are rather complicated, and in [4], only an example was considered. In the present paper, we compute the principal term of the asymptotics and a remainder estimate, under simple and rather natural conditions on (slowly growing) q.…”