Discrete spectrum in the zero-core limit

Determine whether taking the hard-core radius to zero yields a discrete energy spectrum for the attractive inverse-power potential under the Dirichlet boundary condition u(r)=0 at r=0, despite quantum collapse driving the ground-state energy to minus infinity.

Background

The paper studies bound states of the Schrödinger equation with an attractive inverse-power potential U(r)=-U₀/(2rⁿ), whose singularity at the origin produces quantum collapse for appropriate parameters. Without a microscopic boundary condition at r=0, the singular bound states form a continuous energy spectrum because the boundary condition remains loose.

Introducing an impenetrable hard core of radius r₀ and imposing the Dirichlet condition u(r)=0 for r≤r₀ regularizes the singularity and produces a discrete spectrum, including a ground state. However, the authors report that the ground-state energy tends to negative infinity as r₀ approaches zero in the collapsing regime, and they leave unresolved whether discreteness survives in the limiting problem with the Dirichlet condition imposed directly at the origin.