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

TL;DR

This paper establishes Weyl laws for counting exponentially small singular values of the semiclassical  operator on a torus, using optimal weights derived from free boundary problems to obtain precise asymptotics.

Contribution

It introduces a method to derive Weyl asymptotics for small singular values of the  operator using optimal exponential weights from free boundary problems.

Findings

Derived upper and lower bounds for the number of small singular values.

Constructed optimal weights solving a free boundary problem.

Provided precise Weyl asymptotics for the singular value counting function.

Abstract

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 $[0,\mathrm{e}^{-\tau/h}]$, for $\tau>0$ 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 $\tau > 0$.