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

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 (Formula presented.) 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, (Formula presented.). The sectoral, tesseral, and zonal modes, which correspond to (Formula presented.), (Formula presented.), and (Formula presented.), respectively, are found in an approximate analytical form for relatively small values of (Formula presented.). Explicit results are produced for the p- and d-wave states, with (Formula presented.) and 2, respectively. In the general form, the bound states are obtained numerically, confirming the accuracy of the analytical approximation.