Laplace Operator - Spectral Theory

Spectral Theory

The spectrum of the Laplace operator consists of all eigenvalues λ for which there is a corresponding eigenfunction ƒ with

This is known as the Helmholtz equation. If Ω is a bounded domain in Rn then the eigenfunctions of the Laplacian are an orthonormal basis for the Hilbert space L2(Ω). This result essentially follows from the spectral theorem on compact self-adjoint operators, applied to the inverse of the Laplacian (which is compact, by the Poincaré inequality and Kondrakov embedding theorem). It can also be shown that the eigenfunctions are infinitely differentiable functions. More generally, these results hold for the Laplace–Beltrami operator on any compact Riemannian manifold with boundary, or indeed for the Dirichlet eigenvalue problem of any elliptic operator with smooth coefficients on a bounded domain. When Ω is the n-sphere, the eigenfunctions of the Laplacian are the well-known spherical harmonics.