Eigenvalue inequalities for Klein–Gordon operators

We consider the pseudodifferential operators Hm,ΩHm,Ω associated by the prescriptions of quantum mechanics to the Klein–Gordon Hamiltonian |P|2+m2 when restricted to a bounded, open domain Ω∈RdΩ∈Rd. When the mass m   is 0 the operator H0,ΩH0,Ω coincides with the generator of the Cauchy stochastic process with a killing condition on ∂Ω  . (The operator H0,ΩH0,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⩽⋯0<β1<β2⩽⋯ of Hm,ΩHm,Ω and their means βk¯:=1k∑ℓ=1kβℓ.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⩾2d⩾2 and any k,βk+1⩽d+1d−1βk¯. Furthermore, when d⩾2d⩾2 and k⩾2jk⩾2j,β¯kβ¯j⩽d21/d(d−1)(kj)1d. Finally, we present some analogous estimates allowing for an operator including an external potential energy field, i.e., Hm,Ω+V(x)Hm,Ω+V(x), for V(x)V(x) in certain function classes.