A Hamilton system with a few local minima in the potential energy surface represents a model in frame of which one can describe the dynamics of transitions between different equilibrium states, including such important transitions as chemical reactions, nuclear fission and others. Systems of such type allow existence of several critical energy values even for a fixed set of potential parameters. That leads to a possibility of existence of so-called mixed state [1] for such potentials: different dynamic regimes (regular or stochastic) are realised in the same energy interval in different local minima. It gives new possibilities for studies of the quantum manifestations of classical stochasticity (QMCS), using the following objects: statistical properties of energy spectrum (nearest-neighbour spacing distribution), structure of the wave functions (nodal curves, probability density) and wave packet dynamics. We consider such possibilities in application to quadrupole surface oscillations of nuclei, described by the lowest terms of the deformation potential decomposition in deformation parameters. In that potential the mixed state is observed in that part of the parameter space, where the equilibrium shape of the nucleus can be either spherically symmetric or deformed, i.e. for the potentials with a few local minima. In the case of the potential energy surface of complicated topology numerical calculation based on matrix diagonalization becomes ineffective, but the so-called spectral method [2] can become an inspiring alternative. Since the spectral method is fundamentally based on numerical solutions of a time-dependent differential equation, its implementation is always straightforward. Neither special ad hoc selection of basis function is required, nor is it necessary for the potential to have a special analytic form. The spectral method is in principle applicable to problems involving any number of dimensions.
References
[1] Yu. L. Bolotin, V. Yu. Gonchar, E. V. Inopin, V. V. Levenko,
V. N. Tarasov and N. A. Chekanov, Fiz. Elem. Chastits and At. Yadra 20
(1989) 878
[2] M. D. Feit, J. A. Fleck, Jr., and A. Steiger Journal of
Computational Physics 47 (1982) 412