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One can hear the area of a torus by hearing the eigenvalues of the polyharmonic operators

Guo S., Jin J.

Demonstratio Mathematica

2014

Vol. 47, nr 3

607--614

This 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 λ, we establish a Weyl-type asymptotic formula for the spectral problem of the polyharmonic operators on T, that is, as (…), where (…) is the volume of the unit ball in Rn and Vol T is the area of T, which gives the information of the area of the torus based on the spectrum of the polyharmonic operators.