Abstract:We study the number of exponentially small singular values of the semiclassical $\overline{\partial}$ operator on exponentially weighted $L^2$ spaces on a compact Riemann surface. Accurate upper and lower bounds on the number of such singular values are established in terms 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 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 $\tau > 0$.
From: Martin Vogel [view email]
[v1]
Mon, 12 May 2025 07:25:34 UTC (73 KB)
[v2]
Tue, 27 May 2025 03:06:34 UTC (74 KB)
[v3]
Tue, 23 Jun 2026 19:47:43 UTC (814 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。