Analytic birth--death processes: A Hilbert-space approach

Methods of Hilbert space theory together with the theory of analytic semigroups lead to an alternative approach for discussing an analytic birth and death process with the backward equations gk = [lambda]k - 1gk - 1 - ([mu]k+[lambda]k)gk+[mu]k+1, k = 0, 1, 2, ..., where [lambda]- 1 = 0 = [mu]0. For rational growing forward and backward transition rates [lambda]k = O(k[gamma]), [mu]k = O(k[gamma]) (as k --> [infinity]), with 0 < [gamma] < 1, the existence and uniqueness of a solution (which is analytic for t> 0) can be proved under fairly general conditions; so can the discreteness of the spectrum. Even in the critical case of asymptotically symmetric transition rates [lambda]k ~ [mu]k ~ k[gamma] one obtains for rational growing transition rates with 0 < [gamma] < 1 discreteness of the spectrum, generalizing a result of Chihara (1987) and disproving the traditional belief in a continuous spectrum.