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## Pointwise Weyl Laws for Schrodinger operators with singular potentials

### Cheng Zhang

We consider the Schr\"odinger operators $H_V=-\Delta_g+V$ with singular potentials $V$ on general $n$-dimensional Riemannian manifolds and study whether various forms of pointwise Weyl law remain valid under this perturbation. First, 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. This generalizes the 3-dimensional results by Frank-Sabin to any dimensions. Second, we show that the pointwise Weyl law with the standard sharp error term $O(\lambda^{n-1})$ holds for potentials in $L^n(M)$. This extends the classical results of Avakumovi\'c, Levitan and H\"ormander by obtaining the same error term in the pointwise Weyl laws for the Schr\"odinger operators with rough potentials. Both of the results are expected be sharp by the examples constructed in Frank-Sabin.