Quantum tunneling of a singular potential

TL;DR

This paper investigates quantum tunneling in singular potentials without regularization, revealing unique behaviors such as finite transparency at zero energy and oscillatory tunneling in Coulomb potentials, challenging traditional approaches.

Contribution

It demonstrates how the original singular nature of potentials affects quantum tunneling, providing new insights into the behavior of tunneling in singular potentials without regularization.

Findings

Mildly singular potentials have finite, unusual tunneling transparency.

Coulomb potential exhibits infinite oscillations at zero energy.

Tunneling becomes forbidden in more singular potentials.

Abstract

Singularity of the potential function makes quantum tunneling problem mathematically underdetermined. To circumvent the difficulties it introduced in physics, a potential singularity cutoff is often used, followed by a reverse limit transition, or is a suitable self-adjoint extension of the Hamiltonian along the entire coordinate axis made. However, both of them somehow affect the singular nature of the problem, and so I discuss here how quantum tunneling will behave if the original singular nature of the Schrodinger equation left untuched. To do this, I use the property of the probability density current that the singularities are mutually destroyed in it. It is found that the mildly singular potential has a finite, but unusual tunneling transparency, in particular, a non-zero value at zero energy of incident particle. The tunneling of one dimensional Coulomb potential exhibits infinitely fast and complete oscillation at the zero energy boundary and a suppresion to zero in the high-energy limit. In the more singular region, the tunneling becomes forbidden, theby repeating the well-known result of the regularized counterparts.