Semiclassical scar on tori in high dimension

TL;DR

This paper demonstrates semiclassical scar phenomena on high-dimensional tori for certain elliptic operators, under the sigma-Bruno-Rüssmann condition, revealing positive measure concentration on KAM tori.

Contribution

It introduces the occurrence of semiclassical scars on high-dimensional tori under the sigma-Bruno-Rüssmann condition, expanding understanding beyond the Diophantine case.

Findings

Existence of semiclassical measures with positive mass on KAM tori.

Construction of quasimodes in the semiclassical limit.

Extension of scar phenomena to sigma-Bruno-Rüssmann condition.

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 $\sigma$-Bruno-R\"{u}ssmann condition, instead of the Diophantine one. Its equivalence is described as: for almost all perturbed Hamiltonian's KAM Lagrangian tori $\Lambda_{\omega}$, there exists a semiclassical measure with positive mass on $\Lambda_{\omega}$. 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 $\sigma$-Bruno-R\"{u}ssmann condition.