One Can Hear the Area of a Torus by Hearing the Eigenvalues of the Polyharmonic Operators

Abstract

AbstractThis paper considers the asymptotic properties for the spectrum of a positive integer power l of the Laplace-Beltrami operator acting on an n-dimensional torus T. If N(λ) is the number of eigenvalues counted with multiplicity, smaller than a real positive number, we establish a Weyl-type asymptotic formula for the spectral problem of the polyharmonic operators on T, that is, as λ → +∞N (λ) ~ ωwhere ω