IWOTA 2020 Lancaster

Special session at IWOTA 2020 Lancaster: Spectral theory and differential operators, see the conference page here.


We analyse discrete Schrödinger operators \(H_{\lambda,\alpha,\theta} : \ell^p(\mathbb{Z}) \rightarrow \ell^p(\mathbb{Z})\), \(p \in [1, \infty] \), with Sturmian potential, namely, $$ (H_{\lambda,\alpha,\theta} x)_n = x_{n+1} + x_{n-1} + \lambda v_{\alpha,\theta}(n) x_n ~ , \quad n \in \mathbb{Z} ~, $$ where $$ v_{\alpha,\theta} (n) = \chi_{[1-\alpha,1)}(n\alpha+\theta \mod 1) $$ with coupling constant \(\lambda\in\mathbb{R}\), irrational slope \(\alpha\in [0,1]\) and \(\theta \in [0,1)\). The already mentioned Fibonacci Hamiltonian arises when choosing \(\alpha=\frac 12(\sqrt 5-1)\).

We introduce the finite section method, which is often used to solve operator equations approximately, and apply it first to periodic Schrödinger operators. It turns out that the applicability of the method is always guaranteed for integer-valued potentials provided that the operator is invertible. By using periodic approximations, we find a necessary and sufficient condition for the applicability of the finite section method for aperiodic Schrödinger operators and a numerical method to check it. This talk is based on https://arxiv.org/abs/2104.00711.

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