Uniform spectral asymptotics for semiclassical wells on phase space loops

Alix Deleporte; San Vũ Ng\d{o}c

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arXiv · 2002.00234

Uniform spectral asymptotics for semiclassical wells on phase space loops

Alix Deleporte·San Vũ Ng\d{o}c

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Abstract

We consider semiclassical self-adjoint operators whose symbol, defined on a two-dimensional symplectic manifold, reaches a non-degenerate minimum $b_0$ on a closed curve. We derive a classical and quantum normal form which allows us, in addition to the complete integrability of the system, to obtain eigenvalue asymptotics in a window $(-\infty,b_0+\epsilon]$ for $\epsilon > 0$ independent on the semiclassical parameter. These asymptotics are obtained in two complementary settings: either a symmetry of the system under translation along the curve, or a Morse hypothesis reminiscent of Helffer-Sj\"ostrand's "miniwell" situation.

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Alix Deleporte, San Vũ Ng\d{o}c. 2020-02-01. Uniform spectral asymptotics for semiclassical wells on phase space loops. https://arxiv.org/abs/2002.00234

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