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

## TL;DR

This paper establishes a two-term spectral asymptotic expansion for the eigenvalues of the Dirichlet pseudo-relativistic operator on bounded domains, extending previous results for fractional Laplacians and heat trace asymptotics.

## Contribution

It proves a two-term asymptotic expansion for the eigenvalue counting function of the Dirichlet pseudo-relativistic operator, generalizing prior work for fractional Laplacians.

## Key findings

- Two-term asymptotic expansion for eigenvalues established
- Extension of results from fractional Laplacian to pseudo-relativistic operator
- Improved understanding of heat trace small-time asymptotics

## 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.