Finite Sections of Periodic Schrödinger Operators

We study discrete Schrödinger operators $H$ with periodic potentials as they are typically used to approximate aperiodic Schrödinger 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 $\{0, λ\}$-valued potentials with fixed rational $λ$ and period less than nine as well as for arbitrary real-valued potentials of period two.

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