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arXiv · 2502.11445

Semiclassical scar on tori in high dimension

Huanhuan Yuan·Yong Li
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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.

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Huanhuan Yuan, Yong Li. 2025-02-17. Semiclassical scar on tori in high dimension. https://arxiv.org/abs/2502.11445

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