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 $N(\lambda )=\sum _{n=1}^\infty (\lambda _n-\lambda )^{0}_{\_\_}$ of the eigenvalues of the Dirichlet Laplacian (Weyl in Math Ann 71(4):441–479, 1912) and the much later found two-term expansion on domains with highly regular boundary by Ivriĭ (Funktsional Anal i Prilozhen 14(2):25–34, 1980) and Melrose (in: Proceedings of symposia in pure mathematics, vol XXXVI, pp 257–274, American Mathematical Society, 1980), we prove a two-term asymptotic expansion of the Nth Cesàro mean of the eigenvalues of $A_m^\Omega {:}{=} \sqrt{-\Delta + m^2} - m$ for $m>0$ with Dirichlet boundary condition on a bounded domain $\Omega \subset \mathbb R^d$ for $d\ge 2$, extending a result by Frank and Geisinger (J Reine Angew Math 712:1–38, 2016) 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 Bañuelos et al. (J Math Anal Appl 410(2):837–846, 2014) and Park and Song (Potential Anal 41(4):1273–1291, 2014).