Mathematische Zeitschrift Continuity of eigenvalues of subordinate processes in domains

Let X = {Xt, t ≥ 0} be a symmetric Markov process in a state space E and D an open set of E. Let S = {S t , t ≥ 0} be a subordinator with Laplace exponent φn and S = {St , t ≥ 0} a subordinator with Laplace exponent φ. Suppose that X is independent of S and S. In this paper we consider the subordinate processes Xn := {X S (n) t , t ≥ 0} and X := {XSt , t ≥ 0}, and their subprocesses Xn and X killed upon leaving D. Suppose that the spectra of the semigroups of Xn and X are all discrete, with {−λn k ; k ≥ 1} being the eigenvalues of the generator of Xn and {−λ k ; k ≥ 1} being the eigenvalues of the generator of X . We show that, if limn→∞ φn(λ) = φ(λ) for every λ > 0, then lim n→∞ λ φn,D k = λ k for all k ≥ 1. Mathematics Subject Classification (2000): Primary: 58C60, 60J45; Secondary: 35P15, 60G51, 31C25