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Semiclassical scar on tori in high dimension

Huanhuan Yuan, Yong Li

TL;DR

This work extends semiclassical scar phenomena to self-adjoint elliptic operators on the $d$-torus under the $oldsymbol{\sigma}$-Bruno-Rüssmann condition, showing that almost all perturbations retain semiclassical measures with positive mass on KAM tori. The authors construct a quantum Birkhoff normal form in Gevrey regularity, derive a $t$-dependent integrable part $K_{0}(I;t)$, and build a robust family of quasimodes whose mass concentrates on invariant tori. By combining precise separation of quasieigenvalues with lower bounds on overlaps, they prove that eigenfunctions exhibit scars on KAM tori for almost every time parameter $t$ and almost every torus, thereby generalizing prior two-dimensional results to higher dimensions and relaxing resonance conditions. The results provide a rigorous link between classical integrable perturbations, KAM theory in the Gevrey setting, and semiclassical quantum limits, with potential implications for quantum control on high-dimensional manifolds.

Abstract

We show that the eigenfunctions of the self-adjoint elliptic $h-$differential operator $P_{h}(t)$ exhibits semiclassical scar phenomena on the $d-$dimensional torus, under the $σ$-Bruno-Rüssmann condition, instead of the Diophantine one. Its equivalence is described as: for almost all perturbed Hamiltonian's KAM Lagrangian tori $Λ_ω$, there exists a semiclassical measure with positive mass on $Λ_ω$. The premise is that we can obatain a family of quasimodes for the $h-$differential operator $P_{h}(t)$ in the semiclassical limit as $h\rightarrow0$, under the $σ$-Bruno-Rüssmann condition.

Semiclassical scar on tori in high dimension

TL;DR

This work extends semiclassical scar phenomena to self-adjoint elliptic operators on the -torus under the -Bruno-Rüssmann condition, showing that almost all perturbations retain semiclassical measures with positive mass on KAM tori. The authors construct a quantum Birkhoff normal form in Gevrey regularity, derive a -dependent integrable part , and build a robust family of quasimodes whose mass concentrates on invariant tori. By combining precise separation of quasieigenvalues with lower bounds on overlaps, they prove that eigenfunctions exhibit scars on KAM tori for almost every time parameter and almost every torus, thereby generalizing prior two-dimensional results to higher dimensions and relaxing resonance conditions. The results provide a rigorous link between classical integrable perturbations, KAM theory in the Gevrey setting, and semiclassical quantum limits, with potential implications for quantum control on high-dimensional manifolds.

Abstract

We show that the eigenfunctions of the self-adjoint elliptic differential operator exhibits semiclassical scar phenomena on the dimensional torus, under the -Bruno-Rüssmann condition, instead of the Diophantine one. Its equivalence is described as: for almost all perturbed Hamiltonian's KAM Lagrangian tori , there exists a semiclassical measure with positive mass on . The premise is that we can obatain a family of quasimodes for the differential operator in the semiclassical limit as , under the -Bruno-Rüssmann condition.

Paper Structure

This paper contains 23 sections, 168 equations.

Table of Contents

  1. Introduction
  2. Introduction to Semiclassical scar
  3. Regularity assumptions and Non-resonance conditions
  4. Main result
  5. Structure of the work
  6. Preliminaries
  7. Quasimodes
  8. Quantum Birkhoff normal form
  9. The expression of $K_{0}(I;t)$
  10. The classical perturbation theory under --Bruno-Rüssmann condition
  11. Classical theory based on the Hamilton-Jacobi Method
  12. The estimate of the remainder
  13. Calculation of $K_{0}(I;t)$
  14. Semiclassical scarring
  15. Separation of quasieigenvalue
  16. ...and 8 more sections

Theorems & Definitions (8)

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