Editing Laplace operator (section)

==Spectral theory==
{{see also|Hearing the shape of a drum|Dirichlet eigenvalue}}
The [[spectral theory|spectrum]] of the Laplace operator consists of all [[eigenvalue]]s {{math|''λ''}} for which there is a corresponding [[eigenfunction]] {{math|''f''}} with:
<math display="block">-\Delta f = \lambda f.</math>

This is known as the [[Helmholtz equation]].

If {{math|Ω}} is a bounded domain in {{math|'''R'''<sup>''n''</sup>}}, then the eigenfunctions of the Laplacian are an [[orthonormal basis]] for the [[Hilbert space]] {{math|[[Lp space|''L''<sup>2</sup>(Ω)]]}}. This result essentially follows from the [[spectral theorem]] on [[compact operator|compact]] [[self-adjoint operator]]s, applied to the inverse of the Laplacian (which is compact, by the [[Poincaré inequality]] and the [[Rellich–Kondrachov theorem]]).<ref>{{harvnb|Gilbarg|Trudinger|2001|loc=Theorem 8.6}}</ref> It can also be shown that the eigenfunctions are [[infinitely differentiable]] functions.<ref>{{harvnb|Gilbarg|Trudinger|2001|loc=Corollary 8.11}}</ref> 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 {{math|Ω}} is the [[N-sphere|{{mvar|n}}-sphere]], the eigenfunctions of the Laplacian are the [[spherical harmonics]].