Nonlinear Schrödinger equation with power-law confining potential

Abstract

Critical limits of a stationary nonlinear three-dimensional Schr\"odinger equation with confining power-law potentials $(\ensuremath{\sim}{r}^{\ensuremath{\alpha}})$ are obtained using spherical symmetry. When the nonlinearity is given by an attractive two-body interaction (negative cubic term), it is shown how the maximum number of particles ${N}_{c}$ in the trap increases as $\ensuremath{\alpha}$ decreases. With a negative cubic and positive quintic terms we study a first order phase transition, that occurs if the strength ${g}_{3}$ of the quintic term is less than a critical value ${g}_{3c}$. At the phase transition, the behavior of ${g}_{3c}$ with respect to $\ensuremath{\alpha}$ is given by ${g}_{3c}\ensuremath{\sim}0.0036+0.0251∕\ensuremath{\alpha}+0.0088∕{\ensuremath{\alpha}}^{2}$.