One can hear the area of a torus by hearing the eigenvalues of the polyharmonic operators. (English) Zbl 1298.35126
Summary: 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 \(N(\lambda)\) 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 \(\lambda \to +\infty\)
\[
N(\lambda)\sim\omega_n (\mathrm{Vol}T)\lambda^{n/2l}/2(\pi)^n,
\]
where \(\omega_n\) is the volume of the unit ball in \(\mathbb{R}^n\) and \(\mathrm{Vol}T\) is the area of \(T\), which gives the information of the area of the torus based on the spectrum of the polyharmonic operators.
MSC:
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 35R01 | PDEs on manifolds |
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
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