This article is cited in 4 papers

Asymptotics of the distribution of eigenvalues of a selfadjoint second order hyperbolic differential operator on the two-dimensional torus

V. M. Kaplitskiĭ<sup>ab

a</sup> Southern Federal University, Rostov-on-Don, Russia
^b South Mathematical Institute, Vladikavkaz, Russia

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.

Keywords: hyperbolic operator, distribution of eigenvalues, spectrum.

UDC: 517.984.56

Received: 01.10.2009

 English version: Siberian Mathematical Journal, 2010, 51:5, 830–846

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