IL NUOV0 CIMENTO VOT-. 94 B, N. 1 11 Luglio 1986 Harmonic Oscillator with Complex Frequency (*). A. JANNUSSIS Department o] Physics, University oJ Patfas - Patras, Greece Institute ]or Basic Research, 96 Preccott Str.- Cambridge, MA 02138 E. S~URAS Department o/ Physics, University o] Patras . Patras, Greece (ricevuto 1'11 Novembre 1985; manoscritto revisionato ricevuto il 12 Maggio 1986) Summary. -- Ia the present paper we study the piobIem of the harmonic oscillator with complex frequency. A special case of this problem is the determination of the eigenvalues and eigenfunctions of the squeeze oper- ator in quantum optics. The Hamilton operator of the complex harmonic oscillator is non-Hermitian and its study leads to the Lie-admissible theory. Because of the complex trequency the eigcnvalues of the energy are complex numbers and the partition function of Boltzman and the free energy of Helmholtz are complex functions. Especially the imaginary part of the free energy describes the metastable states. PACS. 03.65. - Quantum theory; quantum mechanics. 1. - Introduction. For the case of complex frequencies the Planck's law is written as (1.~) E = ~(~o~ + i~>~). If the eigenvalues of the energy of a physical system are complex numbers, then we accept that the Hamilton operator of the system is non-Hermitian: (1.2) H ---=Ho @ ill1 (*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. 29