  We show that the spacing between eigenvalues of the discrete 1D Hamiltonian
with arbitrary potentials which are bounded, and with Dirichlet or Neumann
Boundary Conditions is bounded away from zero. We prove an explicit lower
bound, given by $Ce^{-bN}$, where $N$ is the lattice size, and $C$ and $b$ are
some finite constants. In particular, the spectra of such Hamiltonians have no
degenerate eigenvalues. As applications we show that to leading order in the
coupling, the solution of a nonlinearly perturbed Anderson model in
one-dimension (on the lattice) remains exponentially localized, in probability
and average sense for initial conditions given by a unique eigenfunction of the
linear problem. We also bound the derivative of the eigenfunctions of the
linear Anderson model with respect to a potential change.
