  Schrodinger eigenproblems on a discrete interval are further investigated
with special attention given to test cases such as the linear and Rosen--Morse
potentials. In the former case it is shown that the characteristic function
determining the eigenvalues is a Lommel polynomial and considerable space is
devoted to these objects. For example it is shown that the continuum limit of
the determinant is obtained by a transitional limit of the Lommel polynomials
for large order and argument.
  Numerical comparisons between discrete approximations and (known) continuum
values for the ratio of functional determinants with and without the potential
are made and show good agreement, even for small numbers of vertices. The zero
mode problem is also briefly dealt with.
