Two-Term Spectral Asymptotics for the Dirichlet Pseudo-Relativistic Kinetic Energy Operator on a Bounded Domain

Abstract

Continuing the series of works following Weyl's one-term asymptotic formula for the counting function $N(\lambda)=\sum_{n=1}^\infty(\lambda_n{-}\lambda)_-$ of the eigenvalues of the Dirichlet Laplacian and the much later found two-term expansion on domains with highly regular boundary by Ivrii and Melrose, we prove a two-term asymptotic expansion of the $N$-th Ces\`aro mean of the eigenvalues of $\sqrt{-\Delta + m^2} - m$ for $m>0$ with Dirichlet boundary condition on a bounded domain $\Omega\subset\mathbb R^d$ for $d\geq 2$, extending a result by Frank and Geisinger for the fractional Laplacian ($m=0$) and improving upon the small-time asymptotics of the heat trace $Z(t) = \sum_{n=1}^\infty e^{-t \lambda_n}$ by Ba\~nuelos et al. and Park and Song.