Heat Kernel Approach to the Weyl Law for Schrödinger Operators on Non-compact Manifolds

Abstract: In a previous work by the authors, the heat kernel expansion for Schr\"odinger-type operators on noncompact manifolds was studied. Motivated by that work, this paper studies Weyl's law and the semiclassical Weyl's law for Schr\"odinger operators of the form $\Delta + V$ and $\hbar^2\Delta + V$ on noncompact complete manifolds with bounded geometry. We assume that the potential $V(x)$ satisfies certain H\"older continuity conditions and tends to $+\infty$ as $x$ approaches infinity.