Eigenvalue inequalities for Klein–Gordon operators

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Abstract

We consider the pseudodifferential operators Hm,Ω associated by the prescriptions of quantum mechanics to the Klein–Gordon Hamiltonian |P|2+m2 when restricted to a bounded, open domain ΩRd. When the mass m is 0 the operator H0,Ω coincides with the generator of the Cauchy stochastic process with a killing condition on ∂Ω. (The operator H0,Ω is sometimes called the fractional Laplacian with power 12, cf. [R. Bañuelos, T. Kulczycki, Eigenvalue gaps for the Cauchy process and a Poincaré inequality, J. Funct. Anal. 211 (2) (2004) 355–423; R. Bañuelos, T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal. 234 (2006) 199–225; E. Giere, The fractional Laplacian in applications, http://www.eckhard-giere.de/math/publications/review.pdf].) We prove several universal inequalities for the eigenvalues 0<β1<β2 of Hm,Ω and their means βk¯:=1k=1kβ.

Among the inequalities proved are:βk¯cst.(k|Ω|)1/d for an explicit, optimal “semiclassical” constant depending only on the dimension d. For any dimension d2 and any k,βk+1d+1d1βk¯. Furthermore, when d2 and k2j,β¯kβ¯jd21/d(d1)(kj)1d. Finally, we present some analogous estimates allowing for an operator including an external potential energy field, i.e., Hm,Ω+V(x), for V(x) in certain function classes.

Keywords

Fractional Laplacian
Weyl law
Dirichlet problem
Riesz means
Universal bounds
Cauchy process
Dirac equation
Klein–Gordon equation
Semiclassical
Relativistic particle

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