Avoided level crossings in quasi-exact approach

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.