Pointwise Weyl Laws for Schrödinger operators with singular potentials

Abstract

We consider the Schrödinger operators H V = −∆g + V with singular potentials V on general n-dimensional Riemannian manifolds and study whether various forms of pointwise Weyl law remain valid under this pertubation. We prove that the pointwise Weyl law holds for potentials in the Kato class, which is the minimal assumption to ensure that H V is essentially self-adjoint and bounded from below or has favorable heat kernel bounds. Moreover, we show that the pointwise Weyl law with the standard sharp error term O(λ n−1 ) holds for potentials in L n (M ).

Sharp Pointwise Weyl Laws for Schrödinger Operators with Singular Potentials on Flat Tori

“…In this paper, we show that under a type of singular perturbations one can obtain the pointwise Weyl law with a sharp error term in any dimensions. Moreover, this result verifies the sharpness of the general theorems for the Schrödinger operators H V = −∆g + V in the previous work [16] of the authors, and extends the 3-dimensional results of to any dimensions.…”

“…In this paper, we show that under a type of singular perturbations one can obtain the pointwise Weyl law with a sharp error term in any dimensions. Moreover, this result verifies the sharpness of the general theorems for the Schrödinger operators H V = −∆g + V in the previous work [16] of the authors, and extends the 3-dimensional results of to any dimensions.understanding the kernel of the spectral projection operator λj ≤λ e j (x)e j (y). Indeed, the pointwise Weyl law holds (0.2) λj ≤λ |e j (x)| 2 = (2π) −n ω n λ n + O(λ n−1 ), uniformly in x.…”

“…See also [26], [7]. In the recent work [16], the authors proved the following pointwise Weyl law for H V . See also [7] for similar results on 3-dimensional manifolds.…”