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  "body": "\n## Statement\n\nLet $M$ be a compact connected [[Riemannian manifold|7CBT4SM83Y]] without boundary, with volume measure $dv$ and total volume $V$.\nAssume that the [[geodesic flow|VF43FTDA1H]] on the unit cotangent bundle $S^*M$ is [[ergodic|AZRPHS478B]] for the Liouville measure.\nLet $(\\varphi_j)_{j \\ge 0}$ be an orthonormal basis of $L^2(M)$ of real eigenfunctions of the [[Laplace–Beltrami operator|F4NWZNTRMD]], with eigenvalues $\\lambda_0 \\le \\lambda_1 \\le \\lambda_2 \\le \\cdots$.\nThen there is a set $S \\subseteq \\N$ of density one, that is, with $\\abs{S \\cap \\set{0, \\ldots, N}} / (N + 1) \\to 1$ as $N \\to \\infty$, such that for every continuous function $a$ on $M$\n$$\\lim_{j \\to \\infty, \\ j \\in S} \\int_M a \\, \\varphi_j^2 \\, dv = \\frac{1}{V} \\int_M a \\, dv.$$\n\n## Remarks\n\nThe full theorem is stated in phase space.\nLet $A$ be a pseudodifferential operator of order $0$ with principal symbol $\\sigma_A$, and let $\\bar{\\sigma}_A$ be the average of $\\sigma_A$ over $S^*M$ for the normalised Liouville measure.\nThen $\\ip{A \\varphi_j}{\\varphi_j}$ tends to $\\bar{\\sigma}_A$ as $j \\to \\infty$ in $S$.\nThe version above is the case of multiplication operators.\nThe proof combines Egorov's theorem, which relates the quantum evolution to the geodesic flow, with the local [[Weyl law|2JK9YMZV2B]] and the mean ergodic theorem.\nThese give that the variance $\\frac{1}{N} \\sum_{j \u003c N} \\abs{\\ip{A \\varphi_j}{\\varphi_j} - \\bar{\\sigma}_A}^2$ over the first $N$ eigenfunctions tends to $0$ as $N \\to \\infty$, which yields a density-one subsequence.\nThe result is due to Shnirelman, Zelditch and Colin de Verdière.\nThe ergodicity hypothesis is needed: on the round sphere $S^2$ the geodesic flow is not ergodic, and there are orthonormal bases of eigenfunctions for which the conclusion fails.\nFor example, the normalised real parts of $(x_1 + i x_2)^k$ restricted to $S^2$ are eigenfunctions whose mass concentrates on the equator $x_3 = 0$ as $k \\to \\infty$.\nWhether the whole sequence converges, with no exceptional subsequence, is the [[quantum unique ergodicity conjecture|F3EB7WZFAC]].\n\n## Explanations\n\n### L1 human\n\nEigenfunctions of the Laplacian are the standing waves of a manifold.\nIf a particle sliding freely along geodesics explores the whole manifold evenly, then almost all high-frequency standing waves also spread their mass evenly.\nThis is the meaning of quantum ergodicity: classical chaos forces the quantum waves to equidistribute.\nThe words \"almost all\" matter, since the theorem allows a thin set of exceptions.\n\n### L2 human\n\nThe theorem is the basic bridge between the dynamics of the geodesic flow and the spectral theory of the Laplacian.\nIt is a statement about averages, proved by a variance estimate, and so it cannot rule out a sparse subsequence of eigenfunctions that concentrates.\nNegatively curved compact manifolds satisfy the hypothesis, since their geodesic flows are ergodic.\nPossible limits of the measures $\\varphi_j^2 \\, dv$ along subsequences are called quantum limits, and much later work studies which measures can occur.\n",
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