Sharp Pointwise Weyl Laws for Schrödinger Operators with Singular Potentials on Flat Tori

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 $n\ge 5$. 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 Schrödinger 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.

58 pages. To appear in Communications in Mathematical Physics

References in corpus (1)

• Restriction of toral eigenfunctions to totally geodesic submanifolds

Cited by in corpus (1)

• Sharp Weyl laws with singular potentials