Laplace–Beltrami eigenfunction

Definition

Let $(M,g)$ be a Riemannian manifold. A nonzero smooth function $\psi:M\to\mathbb C$ is a Laplace–Beltrami eigenfunction with eigenvalue $\lambda$ if

-\Delta_g\psi=\lambda\psi,

where $\Delta_g$ is the Laplace–Beltrami operator. The minus sign makes $\lambda\ge0$ on a compact manifold.

Compact case

On a compact connected manifold without boundary, the spectrum is discrete, each eigenspace is finite-dimensional, and the eigenfunctions form an orthonormal basis of $L^2(M)$ . The zero eigenspace consists of constant functions.

High-frequency scale

For large eigenvalue one often writes $\lambda=h^{-2}$ , where $h\to0$ is the semiclassical parameter. The spatial and phase-space distributions of $L^2$ -normalized eigenfunctions are central objects in quantum chaos.

References

Isaac Chavel, Eigenvalues in Riemannian Geometry, Academic Press, 1984. DOI record.