= Laplacian spectrum
{c}
{title2=$\operatorname{Spec}(\Delta)$}
Use the <positive Laplace-Beltrami operator> $\Delta_+=-\operatorname{div}_g\operatorname{grad}_g$. On a <closed manifold> of positive <dimension>, its <self-adjoint> realization has a discrete <spectrum>
$$
0=\lambda_0\leq\lambda_1\leq\lambda_2\leq\cdots,\qquad\lambda_j\longrightarrow\infty,
$$
counted 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> $k\geq1$, represented by sine and cosine modes.
The opposite convention $\Delta=\operatorname{div}_g\operatorname{grad}_g$, 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.
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