Half-line compressions and finite sections of discrete Schrödinger operators with integer-valued potentials

We study 1D discrete Schrödinger operators H with integer-valued potential and show that, (i), invertibility (in fact, even just Fredholmness) of always implies its half-line compression H+ (zero Dirichlet boundary condition, i.e. matrix truncation). In particular, the eigenvalues avoid zero – all other integers. use this result to conclude (ii), finite section method (approximate inversion via growing truncations) is applicable as soon invertible. The same holds for H+.