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We discuss some aspects of the time picture of tunneling for open quantum systems described by non-Hermitian (NH) Hamiltonians. The concept of sojourn time for such systems is introduced in the framework of the biorthonormal formalism. Due to the various definitions of probability density in the non-Hermitian case, we get three different sojourn times, two real and one complex. We consider as model of a dissipative NH system the complex, generalized parametric oscillator, for which we derive the exact expressions of the three sojourn times in terms of the Wei-Norman characteristic functions entering the non-unitary evolution operators. The special case of the inverted Caldirola-Kanai oscillator with complex friction parameter is investigated for an initial extended wavepacket. We also discuss the Landau-Zener-like transitions of the NH parametric oscillator, i.e. the dissipative tunneling through a dynamical barrier due to the perturbative effect of the damping.

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The perturbation and the variational-perturbation methods are applied for calculating the partition function of one-dimensional oscillators with anharmonicity x2n. New formally simple expressions for the free energy and for the Rayleigh–Schrodinger energy corrections are derived. It is shown that the variational-perturbation method overcomes all the deficiencies of the conventional perturbation method. The results of fifth-order numerical calculations for the free energy of the quartic, quartic–sextic, and octic anharmonic oscillator are highly accurate in the whole range of temperatures. PACS Nos.: 03.65.–w, 05.30.–d

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