Avoided level crossings in quasi-exact approach

TL;DR

This paper introduces a quasi-exact method to analyze avoided level crossings in quantum systems, providing analytic potentials and wave functions to better understand quantum relocalization phenomena.

Contribution

It presents a systematic quasi-exact approach to construct analytic potentials and wave functions near avoided crossings, improving understanding of quantum relocalization instabilities.

Findings

Constructed mutually consistent non-polynomial potentials.

Derived non-numerical multi-Gaussian wave functions.

Applied method to one-dimensional critical-instant setup.

Abstract

The existence of quantum tunneling opens the possibility of a sudden spatial relocalization of a system after a minor modification of its parameters. Such a quantum analogue of the Thom's classical catastrophe would manifest itself, experimentally, via a reordering of the maxima of the probability density paralleled by avoided crossings of the energy levels. Any model (described, say, by an analytic potential with several pronounced minima) is difficult to describe near such an instability because the phenomenon is oversensitive to perturbations. A systematic exact (or, better, quasi-exact) construction of the relocalization instants is proposed here. Its application is considered in the one-dimensional critical-instant setup. The approach is shown to yield the mutually consistent non-polynomial analytic potentials together with the related non-numerical multi-Gaussian-shaped wave functions.