Weyl formulae for Schrödinger operators with critically singular potentials
Xiaoqi Huang, Christopher D. Sogge · Communications in Partial Differential Equations · 2021
We obtain generalizations of classical versions of the Weyl formula involving Schrödinger operators HV=−Δg+V(x) on compact boundaryless Riemannian manifolds with critically singular potentials V. In particular, we extend the classical results of Avakumović [1956 Avakumović, V. G. (1956). Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten. Math. Z. 65(1):327–344. DOI: https://doi.org/10.1007/BF01473886.[Crossref] , [Google Scholar]], Levitan [1952 Levitan, B. M. (1952). On the asymptotic behavior of the spectral function of a self-adjoint differential equation of the second order. Izvestiya Akad. Nauk SSSR. Ser. Mat. 16:325–352. [Google Scholar]] and Hörmander [1968 Hörmander, L. (1968). The spectral function of an elliptic operator. Acta Math. 121(0):193–218. DOI: https://doi.org/10.1007/BF02391913.[Crossref] , [Google Scholar]] by obtaining O(λn−1) bounds for the error term in the Weyl formula in the universal case when we assume that V∈L1(M) with the negative part V−=max{0,−V} belongs to the Kato class, K(M), which is the minimal assumption to ensure that HV 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 [1975 Duistermaat, J. J., Guillemin, V. W. (1975). The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29(1):39–79. DOI: https://doi.org/10.1007/BF01405172.[Crossref], [Web of Science ®] , [Google Scholar]] theorem yielding o(λn−1) bounds for the error term under generic conditions on the geodesic flow, and we can also extend Bérard’s (1977 Bérard, P. H. (1977). On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z. 155(3):249–276. DOI: https://doi.org/10.1007/BF02028444.[Crossref], [Web of Science ®] , [Google Scholar]) theorem yielding O(λn−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∈Lp(M) and V−∈K(M) for appropriate exponents p = pn.