Angular-Momentum Modes in a Bosonic Condensate Trapped in the Inverse-Square Potential

Abstract

In the mean-field approximation, the well-known effect of the critical quantum collapse in a 3D gas of particles pulled to the center by potential U(r)=−U0/2r2 is suppressed by repulsive inter-particle interactions, which create the otherwise non-existing s-wave ground state. Here, we address excited bound states carrying angular momentum, with the orbital and magnetic quantum numbers l and m. They exist above a threshold value of the potential’s strength, U0>l(l+1). The sectoral, tesseral, and zonal modes, which correspond to m=l, 0<m<l, and m=0, respectively, are found in an approximate analytical form for relatively small values of U0−l(l+1). Explicit results are produced for the p- and d-wave states, with l=1 and 2, respectively. In the general form, the bound states are obtained numerically, confirming the accuracy of the analytical approximation.