Sharp Pointwise Weyl Laws for Schr\"odinger Operators with Singular Potentials on Flat Tori

TL;DR

This paper proves sharp pointwise Weyl laws with optimal error terms for Schrödinger operators with singular potentials on flat tori across all dimensions, extending previous results and addressing longstanding open problems.

Contribution

It establishes sharp pointwise Weyl laws with optimal error bounds for Schrödinger operators with singular potentials in any dimension, generalizing prior work and solving a key open problem.

Findings

Sharp error term achieved in Weyl law for all dimensions

Extension of Frank-Sabin results to higher dimensions

Verification of sharpness of previous general theorems

Abstract

The Weyl law of the Laplacian on the flat torus $\mathbb{T}^n$ is concerning the number of eigenvalues $\le\lambda^2$, which is equivalent to counting the lattice points inside the ball of radius $\lambda$ in $\mathbb{R}^n$. The leading term in the Weyl law is $c_n\lambda^n$, while the sharp error term $O(\lambda^{n-2})$ is only known in dimension $n\ge5$. Determining the sharp error term in lower dimensions is a famous open problem (e.g. Gauss circle problem). In this paper, we show that under a type of singular perturbations one can obtain the pointwise Weyl law with a sharp error term in any dimensions. Moreover, this result verifies the sharpness of the general theorems for the Schr\"odinger operators $H_V=-\Delta_{g}+V$ in the previous work of the authors, and extends the 3-dimensional results of Frank-Sabin to any dimensions.