We consider the pseudodifferential operators associated by the prescriptions of quantum mechanics to the Klein–Gordon Hamiltonian when restricted to a bounded, open domain . When the mass m is 0 the operator coincides with the generator of the Cauchy stochastic process with a killing condition on ∂Ω. (The operator is sometimes called the fractional Laplacian with power , 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 of and their means .
Among the inequalities proved are: for an explicit, optimal “semiclassical” constant depending only on the dimension d. For any dimension and any k, Furthermore, when and , Finally, we present some analogous estimates allowing for an operator including an external potential energy field, i.e., , for in certain function classes.