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 = −∆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.

Pointwise Weyl Laws for Schrödinger operators with singular potentials

“…In this paper, we prove the pointwise Weyl law on general n-dimensional manifolds, for the Schrödinger operators H V with critically singular potentials V . Our proof extends the wave equation method in [7], [16].…”

“…We give the proof of Lemma 13 and Lemma 12. They are essentially analogous to the lemmas in [7], but we prove them here for the sake of completeness.…”

“…First, we follow the reduction argument in [7]. Let cos tP 0 (x, y) = j cos tλ j e 0 j (x)e 0 j (y).…”

“…Recently, Huang-Sogge [7] proved that if V ∈ K(M ), then the sharp Weyl law of the same form still holds for the Schrödinger operators H V , i.e. (0.9) N V (λ) := #{k :…”

Uniform Sobolev Estimates on compact manifolds involving singular potentials

“…On the other hand, for the exponents arising in uniform Sobolev assumptions we merely need to assume (1.2). There is also recent related work of the second and fourth authors [16] and Frank and Sabin [11] involving the Weyl counting problem for Kato potentials. Using the uniform Sobolev estimates that we shall prove, we shall easily be able to obtain L q quasimode estimates for the optimal range of exponents (1.8), and if we assume, in addition to (1.2), that V − = max{0, −V } is in the Kato space K(M ), we shall also be able to prove quasimode estimates for larger exponents.…”