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, В§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 Schrodinger 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.