Weyl Law for Schrödinger Operators on Noncompact Manifolds, Heat Kernel, and Karamata-Hardy-Littlewood Theorem

Building on our earlier work on heat kernel asymptotics for Schrödinger-type operators on noncompact manifolds, we establish both the classical and semiclassical Weyl laws for Schrödinger operators of the form $Δ+V$ and $\hbar^2Δ+V$ on complete noncompact manifolds. While the semiclassical law can be approached via localization, the classical Weyl law has remained widely expected but unproven in this generality. We impose a mild bounded integral oscillation condition on $ V $ in addition to the assumptions that $V$ diverges at infinity and satisfies a doubling condition. In this setting, our oscillation condition is sharp and strictly weaker than all previously known assumptions, even in the Euclidean case. A central novelty of our approach is an extended Karamata-Hardy-Littlewood Tauberian theorem, adapted to accommodate non-regularly varying spectral asymptotics in noncompact settings, together with its semiclassical analogue. These Tauberian tools allow us to derive both versions of Weyl's law within a unified framework.