Existence and orbital stability of standing waves to a nonlinear Schrodinger equation with inverse square potential on the half-line

In our work, we establish the existence of standing waves to a nonlinear Schrodinger equation with inverse-square potential on the half-line. We apply a profile decomposition argument to overcome the difficulty arising from the non-compactness of the setting. We obtain convergent minimizing sequences by comparing the problem to the problem at "infinity" (i.e., the equation without inverse square potential). Finally, we establish orbital stability/instability of the standing wave solution for mass subcritical and supercritical nonlinearities respectively.