Short Loops and Pointwise Spectral Asymptotics

TL;DR

This paper investigates pointwise semiclassical spectral asymptotics, focusing on how short loops influence the spectral kernel's behavior in low-dimensional Schrödinger operators and near boundaries.

Contribution

It identifies specific cases where short loops significantly affect spectral asymptotics, expanding understanding of spectral behavior in these scenarios.

Findings

Short loops contribute above O(h^{1-d}) in certain low-dimensional cases.

Boundary effects alter spectral asymptotics near edges.

Potential V=0 with non-zero gradient impacts spectral kernel asymptotics.

Abstract

We consider pointwise semiclassical spectral asymptotics i.e. asymptotics of $e(x,x,0)$ as $h\to +0$ where $e(x,y,\tau)$ is the Schwartz kernel of the spectral projector and consider two cases when schort loops give contribution above $O(h^{1-d})$: (i) Schroedinger operator in dimensions $1,2$ as potential $V=0\implies \nabla V\ne 0$; (ii) Operators near boundaries.