Title: 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 $d\geq 2$,
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 Bañuelos et al. and Park and Song.