Half-line compressions and finite sections of discrete Schr\"odinger operators with integer-valued potentials

TL;DR

This paper proves that for 1D discrete Schrödinger operators with integer potentials, invertibility implies the invertibility of their half-line restrictions, ensuring the finite section method's applicability for such operators.

Contribution

It establishes a link between invertibility of the full operator and its half-line restriction for integer-valued potentials, and confirms the finite section method's effectiveness in this setting.

Findings

Invertibility of $H$ implies invertibility of $H_+$.

Dirichlet eigenvalues avoid zero and other integers.

Finite section method applies when $H$ is invertible.

Abstract

We study 1D discrete Schr\"odinger 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 $H_+$.