---
tag: GRC1E284T7
kind: definition
name: Spectrum of the Laplace–Beltrami operator
msc: [58J50, 35P15]
defines: B8TYKV7AA1
uses: [7CBT4SM83Y, F4NWZNTRMD, 0BQAA2BK3Q, A1G25R99PY]
sources:
  - url: https://en.wikipedia.org/wiki/Spectral_geometry
---

## Definition

Let $(M, g)$ be a closed connected [[Riemannian manifold|7CBT4SM83Y]] and let $\Delta = -\operatorname{div} \operatorname{grad}$ be its [[Laplace–Beltrami operator|F4NWZNTRMD]].
A real number $\lambda$ is an [[eigenvalue|0BQAA2BK3Q]] of $\Delta$ if $\Delta \phi = \lambda \phi$ for some smooth function $\phi \ne 0$ on $M$.
The set of eigenvalues is discrete, each eigenspace is finite-dimensional, and all eigenvalues are $\ge 0$.
The *spectrum* of $M$ is the sequence
$$0 = \lambda_0 < \lambda_1 \le \lambda_2 \le \cdots$$
of eigenvalues of $\Delta$, each repeated as many times as the dimension of its eigenspace.

## Remarks

The properties follow from the [[spectral theorem for compact self-adjoint operators|A1G25R99PY]] applied to the inverse of $\Delta + 1$, which is compact on $L^2(M)$ by elliptic regularity and Rellich's theorem.
So $\lambda_j \to \infty$, and the normalized eigenfunctions form an orthonormal basis of $L^2(M)$.
The eigenvalue $0$ has multiplicity $1$ because $M$ is connected: $\int_M \phi \Delta \phi \, dv = \int_M \abs{\nabla \phi}^2 dv$ vanishes only for constant $\phi$.
Some authors use the opposite sign $\Delta = \operatorname{div} \operatorname{grad}$, with eigenvalues $\le 0$.

Example: for the round unit sphere $S^2$, the eigenvalues are $k(k+1)$, $k = 0, 1, 2, \ldots$, with multiplicity $2k + 1$, the eigenfunctions being the spherical harmonics of degree $k$.
So the spectrum begins $0, 2, 2, 2, 6, 6, 6, 6, 6, \ldots$.

Non-example: $\R$ is not compact, and the definition fails there.
The equation $-\phi'' = \lambda \phi$ has bounded solutions $e^{\pm i\sqrt{\lambda} x}$ for every $\lambda \ge 0$, but none of them is in $L^2(\R)$.
The spectrum of the Laplacian on $L^2(\R)$ is the continuous set $[0, \infty)$, not a sequence of eigenvalues.