  In nuclear-spin-polarization-induced quantum dots the electrons are confined
through local nuclear spin polarization. The model electron confinement
potential is time-dependent due to the nuclear spin diffusion and relaxation
processes. It can be well-approximated by a Gaussian curve which is not an
exactly solvable potential. We demonstrate that it can also be approximated by
multisoliton potentials for the zero value of the angular momentum and by their
singular analogues for other values of momentum without any loss of
calculational accuracy. We obtain these potentials by supersymmetric (or
equivalently Darboux) transformations from the zero potential. The main
advantage of such potentials is that they are exactly solvable. Time-dependence
of the nuclear-spin-polarization-induced quantum dot energy levels is found.
