IWOTA 2020 Lancaster

24/8/2021 One-minute read

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

Abstract:

We analyse discrete Schr├Âdinger 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 Schr├Âdinger 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 Schr├Âdinger operators and a numerical method to check it. This talk is based on https://arxiv.org/abs/2104.00711.