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 Cesàro 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 $d\ge 2$, 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 Bañuelos et al. and Park and Song.

Ann. Henri Poincaré (2018)

References in corpus (2)

• Dimensions, Whitney covers, and tubular neighborhoods
• Two-term spectral asymptotics for the Dirichlet pseudo-relativistic kinetic energy operator on a bounded domain

Cited by in corpus (1)

• Two-term spectral asymptotics for the Dirichlet pseudo-relativistic kinetic energy operator on a bounded domain