Math 2.0

Global analysis, analysis on manifolds Spectral geometry Theorem

Quantum ergodicity

Builds on Riemannian manifoldLaplace–Beltrami operatorSpectrum of the Laplace–Beltrami operatorGeodesic flowErgodic transformation

Statement

Let MM be a compact connected Riemannian manifold without boundary, with volume measure dvdv and total volume VV. Assume that the geodesic flow on the unit cotangent bundle S∗MS^*M is ergodic for the Liouville measure. Let (φj)j≥0(\varphi_j)_{j \ge 0} be an orthonormal basis of L2(M)L^2(M) of real eigenfunctions of the Laplace–Beltrami operator, with eigenvalues λ0≤λ1≤λ2≤⋯\lambda_0 \le \lambda_1 \le \lambda_2 \le \cdots. Then there is a set S⊆ℕS \subseteq \N of density one, that is, with ∣S∩{0,…,N}∣/(N+1)→1\abs{S \cap \set{0, \ldots, N}} / (N + 1) \to 1 as N→∞N \to \infty, such that for every continuous function aa on MM lim⁡j→∞, j∈S∫Ma φj2 dv=1V∫Ma dv.\lim_{j \to \infty, \ j \in S} \int_M a \, \varphi_j^2 \, dv = \frac{1}{V} \int_M a \, dv.

Remarks

The full theorem is stated in phase space. Let AA be a pseudodifferential operator of order 00 with principal symbol σA\sigma_A, and let σˉA\bar{\sigma}_A be the average of σA\sigma_A over S∗MS^*M for the normalised Liouville measure. Then ⟨Aφj,φj⟩\ip{A \varphi_j}{\varphi_j} tends to σˉA\bar{\sigma}_A as j→∞j \to \infty in SS. The version above is the case of multiplication operators. The proof combines Egorov's theorem, which relates the quantum evolution to the geodesic flow, with the local Weyl law and the mean ergodic theorem. These give that the variance 1N∑j<N∣⟨Aφj,φj⟩−σˉA∣2\frac{1}{N} \sum_{j < N} \abs{\ip{A \varphi_j}{\varphi_j} - \bar{\sigma}_A}^2 over the first NN eigenfunctions tends to 00 as N→∞N \to \infty, which yields a density-one subsequence. The result is due to Shnirelman, Zelditch and Colin de Verdière. The ergodicity hypothesis is needed: on the round sphere S2S^2 the geodesic flow is not ergodic, and there are orthonormal bases of eigenfunctions for which the conclusion fails. For example, the normalised real parts of (x1+ix2)k(x_1 + i x_2)^k restricted to S2S^2 are eigenfunctions whose mass concentrates on the equator x3=0x_3 = 0 as k→∞k \to \infty. Whether the whole sequence converges, with no exceptional subsequence, is the quantum unique ergodicity conjecture.

Going deeper

For a student

The theorem is the basic bridge between the dynamics of the geodesic flow and the spectral theory of the Laplacian. It is a statement about averages, proved by a variance estimate, and so it cannot rule out a sparse subsequence of eigenfunctions that concentrates. Negatively curved compact manifolds satisfy the hypothesis, since their geodesic flows are ergodic. Possible limits of the measures φj2 dv\varphi_j^2 \, dv along subsequences are called quantum limits, and much later work studies which measures can occur.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Ergodic transformation, concept, no evidence levelErgodic transformationconceptGeodesic, concept, no evidence levelGeodesicconceptLaplace–Beltrami operator, concept, no evidence levelLaplace–Beltrami oper…conceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptRiemannian metric, concept, no evidence levelRiemannian metricconceptSectional curvature, concept, no evidence levelSectional curvatureconceptSmooth function, concept, no evidence levelSmooth functionconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptSpectrum of the Laplace–Beltrami operator, concept, no evidence levelSpectrum of the Lapla…conceptRiemannian manifold, definition, no evidence levelRiemannian manifolddefinitionTangent bundle, definition, no evidence levelTangent bundledefinitionGeodesic flow, example, no evidence levelGeodesic flowexampleQuantum ergodicity, statement, no evidence levelQuantum ergodicitystatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Ergodic transformation concept
  2. Geodesic concept
  3. Laplace–Beltrami operator concept
  4. Riemannian manifold concept
  5. Riemannian metric concept
  6. Sectional curvature concept
  7. Smooth function concept
  8. Smooth manifold concept
  9. Spectrum of the Laplace–Beltrami operator concept
  10. Riemannian manifold definition
  11. Tangent bundle definition
  12. Geodesic flow example
  13. Quantum ergodicity statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Quantum_ergodicity
About this page
TagD9W79NKYZJ
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Kindstatement, theorem
WritingW0 stub
MSC58J51, 81Q50, 37D40
StatusThis copy of the site has no build data. Run m2 status D9W79NKYZJ.
AboutSpectrum of the Laplace–Beltrami operator
UsesRiemannian manifold, Laplace–Beltrami operator, Spectrum of the Laplace–Beltrami operator, Geodesic flow, Ergodic transformation
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