We obtain generalizations of classical versions of the Weyl formula involving Schrödinger operators H V = −∆g + V (x) on compact boundaryless Riemannian manifolds with critically singular potentials V . In particular, we extend the classical results of Avakumović [1], Levitan [13] and Hörmander [8] by obtaining O(λ n−1 ) bounds for the error term in the Weyl formula in the universal case when we merely assume that V belongs to the Kato class, K(M ), which is the minimal assumption to ensure that H V is essentially self-adjoint and bounded from below or has favorable heat kernel bounds. In this case, we can also obtain extensions of the Duistermaat-Guillemin [4] theorem yielding o(λ n−1 ) bounds for the error term under generic conditions on the geodesic flow, and we can also extend Bérard's [2] theorem yielding O(λ n−1 / log λ) error bounds under the assumption that the principal curvatures are non-positive everywhere. We can obtain further improvements for tori, which are essentially optimal, if we strengthen the assumption on the potential to V ∈ L p (M ) ∩ K(M ) for appropriate exponents p = pn.
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