 Laplacian spectrum

Use the positive Laplace-Beltrami operator ${\Delta }_{+}=-{\mathrm{div}}_g{\mathrm{grad}}_g$. On a closed manifold of positive dimension, its self-adjoint realization has a discrete spectrumcounted with the finite dimensions of the Laplacian eigenspaces. The zero eigenspace consists of functions constant on each connected component. On the unit circle, $-d^2/d{\theta }^2$ has eigenvalue zero once and eigenvalues $k^2$ twice for every integer $k\ge 1$, represented by sine and cosine modes.

The opposite convention $\Delta ={\mathrm{div}}_g{\mathrm{grad}}_g$, used in the coordinate formula for the Laplace-Beltrami operator, negates these eigenvalues. Specifying the sign is therefore part of specifying the spectrum.

On a compact manifold with boundary, Dirichlet boundary conditions or Neumann boundary conditions give corresponding discrete self-adjoint spectra. On a noncompact manifold, continuous spectrum can occur. The operator domain and any boundary conditions belong to the spectral problem, rather than being determined by the local differential formula alone.