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. This result establishes the sharpness of the general theorems for the Schrödinger operators $H_V=-\Delta _{g}+V$ in the previous work (Huang and Zhang (Adv Math, arXiv:2103.05531)) of the authors, and extends the 3-dimensional results of Frank and Sabin (Sharp Weyl laws with singular potentials. arXiv:2007.04284) to any dimensions by using a different approach. Our approach is a combination of Fourier analysis techniques on the flat torus, Li–Yau’s heat kernel estimates, Blair–Sire–Sogge’s eigenfunction estimates, and Duhamel’s principle for the wave equation.