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

Michael Hitrik, Johannes Sjöstrand, Martin Vogel

TL;DR

This work establishes Weyl-type asymptotics for the number of exponentially small singular values of the semiclassical $\overline{\partial}$ operator on exponentially weighted spaces over the 2D torus. The authors develop Hörmander-Carleman estimates with superharmonic weights and reduce the counting problem to trace-class Bergman/Töplitz-type projections, yielding precise upper and lower bounds that converge to a Weyl law in the small-$h$ limit. A central contribution is the construction of an optimal weight $\psi$ solving a double obstacle problem on a compact manifold, which delivers the leading term $\frac{1}{2\pi h}\int_{M_+(\psi)}\Delta\varphi$ in the Weyl formula; in the small-$\tau$ regime, they obtain refined thin-band corrections expressed through the geometry of the curve $\gamma = (\Delta\varphi)^{-1}(0)$. The work also connects tunneling phenomena in magnetic-type operators to this setting and provides a general framework for optimal weights via free boundary problems, including existence, regularity, and quadratic growth near free boundaries.

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 $[0,\mathrm{e}^{-τ/h}]$, 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 $τ> 0$.

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

TL;DR

This work establishes Weyl-type asymptotics for the number of exponentially small singular values of the semiclassical operator on exponentially weighted spaces over the 2D torus. The authors develop Hörmander-Carleman estimates with superharmonic weights and reduce the counting problem to trace-class Bergman/Töplitz-type projections, yielding precise upper and lower bounds that converge to a Weyl law in the small- limit. A central contribution is the construction of an optimal weight solving a double obstacle problem on a compact manifold, which delivers the leading term in the Weyl formula; in the small- regime, they obtain refined thin-band corrections expressed through the geometry of the curve . The work also connects tunneling phenomena in magnetic-type operators to this setting and provides a general framework for optimal weights via free boundary problems, including existence, regularity, and quadratic growth near free boundaries.

Abstract

We study the number of exponentially small singular values of the semiclassical operator on exponentially weighted 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 , for 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 .
Paper Structure (26 sections, 30 theorems, 569 equations, 1 figure)

This paper contains 26 sections, 30 theorems, 569 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. The setting
  3. Upper and lower bounds on the number of exponentially small singular values.
  4. Optimal weights and Weyl asymptotics
  5. The case when $\tau$ is small: thin bands of separation
  6. Connection to recent works on tunneling for operators with magnetic fields
  7. Organization of the paper
  8. Hörmander-Carleman estimates
  9. Bounds on the number of exponentially small singular values
  10. Upper bounds
  11. Lower bounds
  12. Singular values for small exponential decay rates: thin bands of separation
  13. Optimal weights via a free boundary problem
  14. Existence, regularity and uniqueness of solutions to the double obstacle problem
  15. The contact sets and the free boundary
  16. ...and 11 more sections

Key Result

Theorem 1

Let $\psi\in C(M;\mathbb R)$ be an upper bound weight, in the sense of Definition def_ubw_n. Assume that the contact set $M_+(\psi)$ is contained in $M_+ = \{z\in M; \Delta \varphi(z) > 0\}$ and that ${\rm int}(M_+(\psi)) \neq \emptyset$. Then the number $N([0,e^{-\tau/h}]$ of singular values of $P$

Figures (1)

  • Figure 1: An illustration of the penalty function $g_\varepsilon(t)$ above \ref{['eq:pen1b0']}.

Theorems & Definitions (50)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • ...and 40 more