Supersymmetry approach to nuclear-spin-polarization-induced quantum dot structure calculations

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.