Eigenvalue inequalities for Klein-Gordon Operators

We consider the pseudodifferential operators $H_{m,\Omega}$ associated by the prescriptions of quantum mechanics to the Klein-Gordon Hamiltonian $\sqrt{|{\bf P}|^2+m^2}$ when restricted to a compact domain $\Omega$ in ${\mathbb R}^d$. When the mass $m$ is 0 the operator $H_{0,\Omega}$ coincides with the generator of the Cauchy stochastic process with a killing condition on $\partial \Omega$. (The operator $H_{0,\Omega}$ is sometimes called the {\it fractional Laplacian} with power 1/2, cf. \cite{Gie}.) We prove several universal inequalities for the eigenvalues $0 < \beta_1 < \beta_2 \le >...$ of $H_{m,\Omega}$ and their means $\overline{\beta_k} := \frac{1}{k} \sum_{\ell=1}^k{\beta_\ell}$. Among the inequalities proved are: {\overline{\beta_k}} \ge {\rm cst.} (\frac{k}{|\Omega|})^{1/d} for an explicit, optimal "semiclassical" constant, and, for any dimension $d \ge 2$ and any $k$: \beta_{k+1} \le \frac{d+1}{d-1} \overline{\beta_k}. Furthermore, when $d \ge 2$ and $k \ge 2j$, \frac{\overline{\beta}_{k}}{\overline{\beta}_{j}} \leq \frac{d}{2^{1/d}(d-1)}(\frac{k}{j})^{\frac{1}{d}}. Finally, we present some analogous estimates allowing for an external potential energy field, i.e, $H_{m,\Omega}+ V(\bf x)$, for $V(\bf x)$ in certain function classes.