Critical stability of particle confined in two- and three-dimensional Gaussian potential

The critical stability of a single particle confined in a Gaussian potential well in both two and three dimensions is investigated. The critical potential parameters at which the system makes a bound-continuum quantum phase transition are obtained for several low-lying bound states by employing the generalized pseudospectral method. Our numerical calculations show that the critical parameters for a three-dimensional Gaussian potential well can be obtained with extremely high accuracy close to the working precision, while the critical parameters in two dimensions converge comparably slower. Such a difference is due to the effective inverse square attractive potential produced by the grand orbital angular momentum in two dimensions. The present results significantly improve the prediction of critical parameters for all bound states in both two and three dimensions compared to previous works, except that non-negligible discrepancies exist in the two-dimensional s-wave states.