  We propose a model for the quantum harmonic oscillator on a discrete lattice
which can be written in supersymmetric form, in contrast with the more direct
discretization of the harmonic oscillator. Its ground state is easily found to
be annihilated by the annihilation operator defined here, and its excitation
spectrum is obtained numerically. The versatility of the model is then used to
calculate, in a simple way, the generalized position-dependent scattering
length for a particle colliding with a single static impurity in a periodic
potential and the exact ground state of an interacting many-body problem in a
one-dimensional ring.
