Boundary conditions in quantum mechanics on the discretized half-line

TL;DR

This paper explores quantum mechanics on a discretized half-line, introducing a family of Hamiltonians with boundary conditions that avoid singularities and connect to continuum limits, with applications to various potentials and black hole models.

Contribution

It constructs a one-parameter family of Hamiltonians on the discretized half-line that generalize Robin boundary conditions and exhibit singularity avoidance mechanisms.

Findings

The spectrum is analyzed using analytic and numerical methods.

The Robin boundary conditions are recoverable in the continuum limit.

Dirichlet boundary condition is generic in the discretized model.

Abstract

We investigate nonrelativistic quantum mechanics on the discretized half-line, constructing a one-parameter family of Hamiltonians that are analogous to the Robin family of boundary conditions in continuum half-line quantum mechanics. For classically singular Hamiltonians, the construction provides a singularity avoidance mechanism that has qualitative similarities with singularity avoidance encountered in loop quantum gravity. Applications include the free particle, the attractive Coulomb potential, the scale invariant potential and a black hole described in terms of the Einstein-Rosen wormhole throat. The spectrum is analyzed by analytic and numerical techniques. In the continuum limit, the full Robin family of boundary conditions can be recovered via a suitable fine-tuning but the Dirichlet-type boundary condition emerges as generic.