Pointwise Weyl Laws for Schr\"odinger operators with singular potentials

We consider the Schr"odinger operators $H_V=-{\mathrm{\Delta }}_g+V$ with singular potentials $V$ on general $n$-dimensional Riemannian manifolds and study whether various forms of pointwise Weyl law remain valid under this pertubation. We prove that the pointwise Weyl law holds for potentials in the Kato class, which is the minimal assumption to ensure that $H_V$ is essentially self-adjoint and bounded from below or has favorable heat kernel bounds. Moreover, we show that the pointwise Weyl law with the standard sharp error term $O({\lambda }^{n-1})$ holds for potentials in $L^n(M)$.