Weyl laws for exponentially small singular values of the $\overline{\partial}$ operator Article Swipe

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Johannes Sjöstrand , Michael Hitrik , Martin Vogel ·

YOU? · · 2025 · Open Access ·

We study the number of exponentially small singular values of the semiclassical $\overline{\partial}$ operator on exponentially weighted $L^2$ spaces on the two-dimensional torus. Accurate upper and lower bounds on the number of such singular values are established with the help of auxiliary notions of upper and lower bound weights. Assuming that the Laplacian of the exponential weight changes sign along a curve, we construct optimal such weights by solving a free boundary problem, which yields a Weyl asymptotics for the counting function of the singular values in an interval of the form $[0, e^{ -\tau /h} ]$, for $τ>0$ smaller than the oscillation of the weight. We also provide a precise description of the leading term in the Weyl asymptotics, in the regime of small $τ>0$.

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Type: preprint
Language: en
Landing Page: https://hal.science/hal-05210387
OA Status: green
OpenAlex ID: https://openalex.org/W4414761095

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Title: Weyl laws for exponentially small singular values of the $\overline{\partial}$ operator

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Type: preprint

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Language: en

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Publication year: 2025

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Publication date: 2025-08-14

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Authors: Johannes Sjöstrand, Michael Hitrik, Martin Vogel

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Landing page: https://hal.science/hal-05210387

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