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

The Weyl law of the Laplacian on flat torus $${\mathbb {T}}^n$$ is concerning number eigenvalues $$\le \lambda ^2$$ , which equivalent to counting lattice points inside ball radius $$\lambda $$ in {R}}^n$$ . leading term $$c_n\lambda ^n$$ while sharp error $$O(\lambda ^{n-2})$$ only known dimension $$n\ge 5$$ Determining lower dimensions a famous open problem (e.g. Gauss circle problem). In this paper, we show that under type singular perturbations one can obtain pointwise with any dimensions. This result establishes sharpness general theorems for Schrödinger operators $$H_V=-\Delta _{g}+V$$ previous work (Huang and Zhang (Adv Math, arXiv:2103.05531 )) authors, extends 3-dimensional results Frank Sabin (Sharp laws potentials. arXiv:2007.04284 ) by using different approach. Our approach combination Fourier analysis techniques torus, Li–Yau’s heat kernel estimates, Blair–Sire–Sogge’s eigenfunction Duhamel’s principle wave equation.