Finite Sections of Periodic Schr\"odinger Operators

TL;DR

This paper investigates the finite section method for periodic Schr"odinger operators, providing criteria for its applicability, especially for operators with various types of periodic potentials, including integer-valued and real-valued cases.

Contribution

It offers an efficient test for the applicability of the finite section method to periodic Schr"odinger operators, extending known results to new classes of potentials.

Findings

Finite section method applicable when $H$ is invertible for integer-valued potentials.

Applicability extends to certain rational-valued potentials with small periods.

Results include cases with arbitrary real-valued potentials of period two.

Abstract

We study discrete Schr\"odinger operators $H$ with periodic potentials as they are typically used to approximate aperiodic Schr\"odinger operators like the Fibonacci Hamiltonian. We prove an efficient test for applicability of the finite section method, a procedure that approximates $H$ by growing finite square submatrices $H_n$. For integer-valued potentials, we show that the finite section method is applicable as soon as $H$ is invertible. This statement remains true for $\{0, \lambda\}$-valued potentials with fixed rational $\lambda$ and period less than nine as well as for arbitrary real-valued potentials of period two.