Eigenvalue inequalities for Klein-Gordon Operators

• Published in: arXiv.org
• Main Authors: Harrell, Evans M, Selma Yildirim Yolcu
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 01.10.2008
• Subjects: Eigenvalues; Inequalities; Operators; Potential energy; Quantum mechanics; Quantum theory; Stochastic processes
• ISSN: 2331-8422

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.