Asymptotics of the Distribution of Eigenvalues of a Selfadjoint Second Order Hyperbolic Differential Operator on the Two-Dimensional Torus

Abstract

We study the asymptotic properties of the discrete spectrum for general selfadjoint second order hyperbolic operators on the two-dimensional torus. For a broad class of operators with sufficiently smooth coefficients and the principal part coinciding with the wave operator in the light cone coordinates we prove the discreteness of the spectrum and obtain an asymptotic formula for the distribution of eigenvalues. In some cases we can indicate the first two asymptotic terms. We discuss the relations of these questions to analytic number theory and mathematical physics.

Weyl asymptotics for tensor products of operators and Dirichlet divisors

“…Reference therein is made to Arnold [3], suggesting to transfer the Weyl formula to hyperbolic equations. The results in [22] can be essentially regarded as a particular case of those from [5]. Expansions of the type (1.5) appear also in the recent paper of Coriasco and Maniccia [10] concerning the spectrum of the so-called SG-operators.…”

“…From the point of view of Mathematical Physics, Kaplitskiȋ [22] has independently studied the spectral properties of operators on the torus T 2 with principal part…”

“…Even, for relevant classes of operators, each point of the lattice corresponds exactly to one of eigenvalues, counted according to the multiplicity, and the computation of N (λ) leads in a natural way to problems of number theory. Let us refer for example to [8,9], [12]- [22], [28]- [32]. In this order of ideas, the attention will be fixed here on operators of the form of tensor products (1.1) P = P 1 ⊗ ... ⊗ P p where the operators P j , j = 1, .…”