Spectral Theory of Differential Operators

Abstract

The spectral theory of operators in a finite-dimensional space first appeared in connection with the description of the frequencies of small vibrations of mechanical systems (see Arnol’d et al. 1985). When the vibrations of a string are considered, there arises a simple eigenvalue problem for a differential operator. In the case of a homogeneous string it suffices to use the classical theory of Fourier series. For an inhomogeneous string it becomes necessary to consider the general Sturm-Liouville problem, which is the eigenvalue problem for a simple one-dimensional differential operator with variable coefficients. Failing to be explicitly soluble, the problem calls for a qualitative and asymptotic study (see Egorov and Shubin 1988a, §9). When considering the vibrations of a membrane or a three-dimensional elastic body, we arrive at the eigenvalue problems for many-dimensional differential operators. Such problems also arise in the theory of shells, hydrodynamics, and other areas of mechanics. One of the richest sources of problems in spectral theory, mostly for Schrödinger operators, is quantum mechanics, in which the eigenvalues of the quantum Hamiltonian, and, more generally, the points of the spectrum of the Hamiltonian, are the possible energy values of the system.