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

08/08/2022

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by   Marko Lindner, et al.

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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_+.