Eigenvalue inequalities for Klein-Gordon Operators

Abstract

We consider the pseudodifferential operators $H_{m,\Omega }$ associated by the prescriptions of quantum mechanics to the Klein-Gordon Hamiltonian $\sqrt{|\mathbf{P}{|}^2+m^2}$ 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{β_k}} \ge {\rm cst.} (\frac{k}{|Ω|})^{1/d} for an explicit, optimal "semiclassical" constant, and, for any dimension $d\ge 2$ and any $k$: β_{k+1} \le \frac{d+1}{d-1} \overline{β_k}. Furthermore, when $d\ge 2$ and $k\ge 2j$, \frac{\overlineβ_{k}}{\overlineβ_{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, $H_{m,\Omega }+V(\mathbf{x})$, for $V(\mathbf{x})$ in certain function classes.

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Cited by in corpus (1)

• Trace identities for commutators, with applications to the distribution of eigenvalues