Weyl laws for Schrödinger operators on compact manifolds with boundary

We prove Weyl laws for Schrödinger operators with critically singular potentials on compact manifolds with boundary. We also improve the Weyl remainder estimates under the condition that the set of all periodic geodesic billiards has measure 0. These extend the classical results by Seeley, Ivrii and Melrose. The proof uses the Gaussian heat kernel bounds for short times and a perturbation argument involving the wave equation.