Structure and dynamics of the cubic anharmonic oscillator

In many areas of atomic, nuclear and solid state physics, the need arises to study the quantum mechanical tunneling of a particle trapped in either a double well or in a metastable potential well. One of the well-known models for such a metastable well is the harmonic oscillator perturbed by a cubic term gx3, where g is the coupling constant. In fact, the mathematical analysis of the cubic anharmonic oscillator is a rather difficult task since the bound states of the (unperturbed) harmonic potential become resonances with a final width under the cubic perturbation. A large number of methods have been proposed for the resonance calculations. In our talk, for example, we show that the standard Rayleigh-Schroedinger perturbation series, if properly resummed by using the generalized Borel-Pade method, may well reproduce not only the position of the resonances but also their widths. An alternative approach, known as dilation transformation method, deals with the complex scaling. Apart from the evaluation of the energy spectrum, this method provides a natural access to generalized eigenfunctions of the cubic oscillator. We demonstrate how the proper set of these eigenfunctions may be utilized for the numerical integration of the time-dependent Schroedinger equation and, hence, for studying the dynamics of the wavepackets in the metastable potential wells. (orig.)