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Continuing the series of works following Weyl's one-term asymptotic formulafor 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-termexpansion on domains with highly regular boundary by Ivrii and Melrose, weprove a two-term asymptotic expansion of the $N$-th Ces\`aro mean of theeigenvalues of $\sqrt{-\Delta + m^2} - m$ for $m>0$ with Dirichlet boundarycondition 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.

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