Asymptotics of the distribution of eigenvalues of a selfadjoint second order hyperbolic differential operator on the two-dimensional torus
The author studies a class of general selfadjoint second order hyperbolic operators \(A\) on the two-dimensional torus \(T^2\) with sufficiently smooth coefficients under some conditions that guarantee that the spectrum is discrete. The method consists in constructing a regularizer for the hyperbolic operator and reducing the problem to studying the spectrum of a linear operator pencil, for which the spectrum of the principal unperturbed part is computed explicitly, and the perturbation does not affect the principal term of the asymptotics of \(N(\lambda ;A):=\)card\(\{j: |\lambda_j(A)|\leq\lambda\}\) as \(\lambda\rightarrow +\infty\). Explicit calculation of the asymptotics for the unperturbed pencil reduces to the classical problem of finding the asymptotics for the number of integer points in the domain bounded by a hyperbola as the domain extends indefinitely.
scientific article; zbMATH DE number 5824086
scientific article