The classical Weyl law for Schr\"odinger operators on complete Riemannian manifolds

TL;DR

This paper provides a new geometric-analytic criterion for the validity of the classical Weyl law for Schr"odinger operators on complete Riemannian manifolds, independent of standard geometric or analytic assumptions.

Contribution

It introduces an intrinsic invariant that characterizes when the Weyl law holds, based on the interplay between geometry, potential growth, and oscillation scale.

Findings

Weyl asymptotics hold if the invariant c_δ(λ) tends to zero as λ approaches infinity.

Explicit examples show the criterion's sharpness and when the Weyl law fails.

The approach does not rely on bounded geometry or doubling conditions on the potential.

Abstract

We establish a criterion for the validity of the classical (non-semiclassical) Weyl law for Schr\"odinger operators $ H=\Delta+V $ on complete Riemannian manifolds. In contrast to existing results, our approach does not rely on standard geometric assumptions such as bounded geometry, nor on analytic assumptions such as the doubling condition on the potential. Instead, we identify a geometric-analytic invariant that encodes the precise balance between the geometry of the manifold, the growth of $V$, and the oscillation scale of $V$. This intrinsic quantity, denoted $c_{\delta}(\lambda)$ admits effective quantitative estimates. We prove that the Weyl asymptotic holds provided $\lim_{\lambda\to\infty} c_\delta(\lambda)=0 .$ The sharpness of this criterion is demonstrated through explicit examples showing that the Weyl law can fail when the criterion is violated.