Singularity resolution by lattice shifts in discretized quantum mechanics
We investigate the robustness of singularity avoidance mechanisms in nonrelativistic quantum mechanics on the discretised real line when lattice points are allowed to approach a singularity of the classical potential. We consider the attractive Coulomb potential and the attractive scale invariant potential, on an equispaced parity-noninvariant lattice and on a non-equispaced parity-invariant lattice, and we examine the energy eigenvalues by a combination of analytic and numerical techniques. While the lowest one or two eigenvalues descend to negative infinity in the singular limit, we find that the higher eigenvalues remain finite and form degenerate pairs, close to the eigenvalues of a theory in which a lattice point at the singularity is regularised either by Thiemann's loop quantum gravity singularity avoidance prescription or by a restriction to the odd parity sector. The approach to degeneracy can be reproduced from a nonsingular discretised half-line quantum theory by tuning a boundary condition parameter. The results show that Thiemann's singularity avoidance prescription and the discretised half-line boundary condition reproduce quantitatively correct features of the singular limit spectrum apart from the lowest few eigenvalues.