Eigenvalue inequalities for Klein–Gordon operators

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 Ω ∈ Rd . 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 := 1 k k =1 β . 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 d 2 and any k, βk+1 d +1 d −1 βk. Furthermore, when d 2 and k 2j , βk βj d 21/d (d −1) k j 1/d .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. © 2009 Elsevier Inc. All rights reserved