  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.
