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

DOI10.1016/J.JMAA.2022.126984arXiv2208.04015MaRDI QIDQ6407272

Marko Lindner, Riko Ukena

Publication date: 8 August 2022

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 $H_{+}$.

Mathematics Subject Classification ID

(Semi-) Fredholm operators; index theories (47A53) Jacobi (tridiagonal) operators (matrices) and generalizations (47B36)

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