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Pointwise Bounds for Joint Eigenfunctions of Quantum Completely Integrable Systems.

Jeffrey Galkowski1, John A Toth2

  • 1Department of Mathematics, University College London, London, UK.

Communications in Mathematical Physics
|July 18, 2020
PubMed
Summary

This study improves pointwise bounds for joint eigenfunctions of quantum completely integrable systems on Riemannian manifolds. The findings offer polynomial gains over standard estimates and exponential decay in specific regions.

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Area of Science:

  • * Mathematical Physics
  • * Differential Geometry
  • * Spectral Theory

Background:

  • * Analysis of eigenfunctions on compact Riemannian manifolds is crucial for understanding quantum systems.
  • * Quantum completely integrable (ACI) systems possess specific properties related to pseudodifferential operators.
  • * Hörmander bounds provide standard estimates for eigenfunctions, but improvements are sought.

Purpose of the Study:

  • * To establish refined pointwise bounds for joint eigenfunctions of ACI systems.
  • * To investigate improvements over existing Hörmander bounds.
  • * To explore exponential decay estimates for eigenfunctions in specific geometric regions.

Main Methods:

  • * Utilizing pseudodifferential operator calculus.
  • * Applying techniques from microlocal analysis.
  • * Developing new estimates for eigenfunctions on Riemannian manifolds.

Main Results:

  • * Polynomial improvements over Hörmander bounds are achieved for typical points.
  • * In dimensions 2 and 3, estimates match the Hardy exponent.
  • * In higher dimensions, a gain of $\frac{n-1}{2n}$ over Hörmander bounds is obtained.
  • * Under real-analyticity, exponential decay estimates are derived for eigenfunctions outside invariant tori projections.

Conclusions:

  • * The study provides significant advancements in understanding eigenfunction localization for ACI systems.
  • * The derived bounds are sharp and offer new insights into the behavior of eigenfunctions.
  • * These results have implications for spectral theory and quantum mechanics on manifolds.