Continuity of eigenvalues of subordinate processes in domains

Let X = {X-t, t >= 0} be a symmetric Markov process in a state space E and D an open set of E. Let S-(n) = {S-t((n)) t >= 0} be a subordinator with Laplace exponent phi(n) and S = {S-t, t >= 0} a subordinator with Laplace exponent phi. Suppose that X is independent of S and S( n). In this paper we consider the subordinate processes X-phi n := {X-St((n)) t >= 0} and X-phi:= {X-St, t >= 0}, and their subprocesses X-phi n,X- D and X-phi,X- D killed upon leaving D. Suppose that the spectra of the semigroups of X-phi n, D and X-phi,X- D are all discrete, with {-lambda(k)(phi n,D) k >= 1} being the eigenvalues of the generator of X-phi n,X-D and {-lambda(k)(phi,D) k >= 1} being the eigenvalues of the generator of X-phi,X-D. We show that, if lim(n-->infinity)phi(n)(lambda) = phi(lambda) for every lambda > 0, then lim(n-->infinity) lambda(k)(phi n,D) = lambda(k)(phi,D) for all k >= 1.