## Abstract

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 {greater-than with dot}_{k} = ?_{k - 1}g_{k - 1} - (µ_{k}+?_{k})g_{k}+µ_{k+1}, k = 0, 1, 2, ..., where ?_{- 1} = 0 = µ_{0}. For rational growing forward and backward transition rates ?_{k} = O(k^{?}), µ_{k} = O(k^{?}) (as k ? 8), with 0 < ? < 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 ?_{k} ~ µ_{k} ~ k^{?} one obtains for rational growing transition rates with 0 < ? < 1 discreteness of the spectrum, generalizing a result of Chihara (1987) and disproving the traditional belief in a continuous spectrum. © 1994.

Original language | English |
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Pages (from-to) | 65-74 |

Number of pages | 10 |

Journal | Stochastic Processes and their Applications |

Volume | 49 |

Issue number | 1 |

Publication status | Published - Jan 1994 |

## Keywords

- analytic semigroups
- discreteness of spectrum
- infinite tridiagonal matrices