Critical potentials and fluctuations phenomena with quartic, sextic, and octic anharmonic oscillator potentials

Abstract

This is an investigation of the energy levels of three classes of two-dimensional anharmonic oscillator potentials characterized by free-, one-, and two- parameters with quartic, sextic, and octic anharmonicity. The wave functions and energy eigenvalues are found through the discretization using the nine-point finite-difference method. It is proved that the free-parameter potentials provide a “bridge” between the two-dimensional harmonic oscillator potential and two-dimensional infinite square well potential. However, the one- and two- parameter potentials show a transition from tunneling splitting mode to the spectra of free-parameter potentials crossing to the spectrum of infinite square well potential and in the same time present new features. One can distinguish cases of fluctuations phenomena in the ground state and excited states. We obtained, for the first time, a ground state with a four-peak structure. The quantum tunneling between the four minima of one- and two- parameter potentials depends on the characteristics of the separating barrier. These phase transitions are not of the ordinary thermodynamic category, but truly they are phase transitions in the structure of the low-lying energy spectra. The concept of critical potentials is associated with the potentials at the phase-transition points. The coherent quadrupole-octupole motion model, using these potentials, is applied to describe yrast and non-yrast energy sequences with alternating parity in several even-even nuclei from different regions, namely, in 100Mo, 146Ba, 226Ra, and 226Th.