Spectrum of the Laplace–Beltrami operator

Summary

On a closed Riemannian manifold the Laplace–Beltrami operator has a discrete spectrum $0 = \lambda_0 < \lambda_1 \le \lambda_2 \le \cdots \to \infty$ of eigenvalues of finite multiplicity. The eigenfunctions are smooth and form an orthonormal basis of $L^2(M)$. This sequence is the basic object of spectral geometry: it controls the heat equation, the wave equation and the vibrations of the manifold.