---
tag: B8TYKV7AA1
kind: concept
name: Spectrum of the Laplace–Beltrami operator
msc: [58J50, 35P15]
sources:
  - url: https://en.wikipedia.org/wiki/Spectral_geometry
---

## Summary

On a closed Riemannian manifold the [[Laplace–Beltrami operator|F4NWZNTRMD]] 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|QKAFRDPGE4]]: it controls the heat equation, the wave equation and the vibrations of the manifold.

## Explanations

### L1 human

Think of the manifold as a drum without boundary.
Its pure tones are the eigenfunctions of the Laplacian, and the squares of their frequencies are the eigenvalues.
The lowest eigenvalue is $0$, for the constant function, which does not vibrate.
On the circle of length $2\pi$ the eigenfunctions are $\cos k\theta$ and $\sin k\theta$, with eigenvalues $k^2$.