  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.
