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.
Categories: math.SP math-ph math.MP