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  "body": "\n## Definition\n\nLet $(M, g)$ be a closed connected [[Riemannian manifold|7CBT4SM83Y]] and let $\\Delta = -\\operatorname{div} \\operatorname{grad}$ be its [[Laplace–Beltrami operator|F4NWZNTRMD]].\nA real number $\\lambda$ is an [[eigenvalue|0BQAA2BK3Q]] of $\\Delta$ if $\\Delta \\phi = \\lambda \\phi$ for some smooth function $\\phi \\ne 0$ on $M$.\nThe set of eigenvalues is discrete, each eigenspace is finite-dimensional, and all eigenvalues are $\\ge 0$.\nThe *spectrum* of $M$ is the sequence\n$$0 = \\lambda_0 \u003c \\lambda_1 \\le \\lambda_2 \\le \\cdots$$\nof eigenvalues of $\\Delta$, each repeated as many times as the dimension of its eigenspace.\n\n## Remarks\n\nThe 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.\nSo $\\lambda_j \\to \\infty$, and the normalized eigenfunctions form an orthonormal basis of $L^2(M)$.\nThe 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 $\\phi$.\nSome authors use the opposite sign $\\Delta = \\operatorname{div} \\operatorname{grad}$, with eigenvalues $\\le 0$.\n\nExample: 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 $k$.\nSo the spectrum begins $0, 2, 2, 2, 6, 6, 6, 6, 6, \\ldots$.\n\nNon-example: $\\R$ is not compact, and the definition fails there.\nThe equation $-\\phi'' = \\lambda \\phi$ has bounded solutions $e^{\\pm i\\sqrt{\\lambda} x}$ for every $\\lambda \\ge 0$, but none of them is in $L^2(\\R)$.\nThe spectrum of the Laplacian on $L^2(\\R)$ is the continuous set $[0, \\infty)$, not a sequence of eigenvalues.\n",
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