Quantum-mechanical wave functions in singular potentials: linear and nonlinear states

This paper investigates the behavior of quantum-mechanical wave functions under the influence of two extreme classes of singular potentials: attractive potentials that exhibit a singularity at the origin (r to 0) and expulsive potentials that grow aggressively at infinity (r to infinity). Traditional quantum mechanics shows that three-dimensional linear Schrödinger equations with an inverse-square potential undergo a catastrophic quantum collapse when the potential strength exceeds a critical threshold. The authors explore how nonlinear terms—specifically repulsive interactions in Bose-Einstein condensates—can stabilize these systems and prevent collapse. Additionally, the authors examine whether steep expulsive potentials, which typically create delocalized continuous states, can paradoxically support normalizable bound states through linear self-trapping.

For the three-dimensional inverse-square potential, the linear Schrödinger equation fails to produce a ground state when the attractive strength exceeds one-quarter. The authors review how incorporating the cubic repulsive term into the Gross-Pitaevskii equation completely suppresses this quantum collapse. By balancing the singular attraction with nonlinear self-repulsion, the system generates a family of stable ground states and angular-momentum-carrying vortex states whose norms remain finite. Similar stabilization mechanisms are demonstrated for two-dimensional systems using quintic self-repulsion to counteract stronger singularities.

Another major focus of the article is the analysis of linear and nonlinear wave equations featuring expulsive potentials that grow faster than a standard harmonic oscillator at large distances. Counter to the intuitive expectation that steep repulsive forces cause extreme delocalization, the linearized equations reveal an unexpected linear self-trapping effect. The asymptotic wave functions oscillate rapidly and decay in a manner that ensures norm convergence across the entire continuous energy spectrum. These normalizable states act as localized bound states embedded within the continuous spectrum, akin to quantum bound states in the continuum.