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

Abstract

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