Weyl formulae for Schrödinger operators with critically singular potentials

Abstract

We obtain generalizations of classical versions of the Weyl formula involving Schrödinger operators $H_V=-{\Delta }_g+V(x)$ on compact boundaryless Riemannian manifolds with critically singular potentials $V$. In particular, we extend the classical results of Avakumović , Levitan and Hörmander by obtaining $O({\lambda }^{n-1})$ bounds for the error term in the Weyl formula in the universal case when we merely assume that $V$ belongs to the Kato class, $\mathcal{K}(M)$, 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 theorem yielding $o({\lambda }^{n-1})$ bounds for the error term under generic conditions on the geodesic flow, and we can also extend Bérard's theorem yielding $O({\lambda }^{n-1}/\log \lambda )$ 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 $V\in L^p(M)\cap \mathcal{K}(M)$ for appropriate exponents $p=p_n$.

A revised version which will appear in Communications in Partial Differential Equations

References in corpus (2)

• Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic
• Sharp Weyl laws with singular potentials

Cited by in corpus (1)

• Sharp Weyl laws with singular potentials