Math 2.0

Global analysis, analysis on manifolds Spectral geometry Conjecture

Quantum unique ergodicity conjecture

Builds on Riemannian manifoldLaplace–Beltrami operatorSpectrum of the Laplace–Beltrami operatorSectional curvatureQuantum ergodicity

Statement

Let MM be a compact connected Riemannian manifold without boundary and with negative sectional curvature, with volume measure dvdv and total volume VV. Let (φj)(\varphi_j) be any sequence of L2L^2-normalised eigenfunctions of the Laplace–Beltrami operator with eigenvalues λj→∞\lambda_j \to \infty. The conjecture says that for every continuous function aa on MM lim⁡j→∞∫Ma ∣φj∣2 dv=1V∫Ma dv.\lim_{j \to \infty} \int_M a \, \abs{\varphi_j}^2 \, dv = \frac{1}{V} \int_M a \, dv. In its full form it says that the microlocal lifts of the measures ∣φj∣2 dv\abs{\varphi_j}^2 \, dv converge to the normalised Liouville measure on the unit cotangent bundle.

Remarks

Quantum ergodicity gives this convergence along a subsequence of density one; the conjecture asks that there be no exceptional subsequence at all. The conjecture is open in general. Lindenstrauss proved the analogue for compact arithmetic hyperbolic surfaces and joint eigenfunctions of the Laplacian and the Hecke operators. Together with work of Soundararajan that rules out escape of mass, this also covers the modular surface. Anantharaman proved that on compact manifolds with Anosov geodesic flow every quantum limit has positive entropy, so eigenfunctions cannot concentrate entirely on a closed geodesic. For some ergodic billiards the analogue fails: Hassell proved that for almost every stadium-shaped billiard there are exceptional sequences of eigenfunctions.

Reading path

To understand this page from the ground up, read these 14 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 ergodicitystatementQuantum unique ergodicity conjecture, conjecture, no evidence levelQuantum unique ergodi…conjecture
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
  14. Quantum unique ergodicity conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Quantum_ergodicity
About this page
TagF3EB7WZFAC
URIm2://F3EB7WZFAC
Kindconjecture
WritingW0 stub
MSC58J51, 81Q50, 11F72
StatusThis copy of the site has no build data. Run m2 status F3EB7WZFAC.
AboutSpectrum of the Laplace–Beltrami operator
UsesRiemannian manifold, Laplace–Beltrami operator, Spectrum of the Laplace–Beltrami operator, Sectional curvature, Quantum ergodicity

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