TUHH / Institut für Mathematik / Forschungsgebiete / Finite Sections of Aperiodic Schrödinger Operators

# Finite Sections of Aperiodic Schrödinger Operators

## Description

Discrete Schrödinger operators are used to describe physical systems on lattices and, therefore, play an important role in theoretical solid-state physics. For a fixed $$p \in [1,\infty]$$, consider the Schrödinger operator $$H \colon \ell^p(\mathbb{Z}) \to \ell^p(\mathbb{Z})$$ given by

$(H x)_n = x_{n + 1} + x_{n - 1} + v(n) x_nn \in \mathbb{Z},$(1)

and its one-sided counterpart $$H_+ \colon \ell^p(\mathbb{N}) \to \ell^p(\mathbb{N})$$ given by

$(H_+ x)_n = x_{n + 1} + x_{n - 1} + v(n) x_n\;,n \in \mathbb{N}, \quad x_0 = 0\;.$ (2)

Based on Definitions $$(1)$$ and $$(2)$$, one can associate $$H$$ and $$H_+$$ with infinite tridiagonal matrices $$A = (a_{ij})_{i,j \in \mathbb{Z}}$$ and $$A_+ = (a_{i,j})_{i,j \in \mathbb{N}}$$.

Looking at the corresponding infinite linear system of equations

$A x = b \quad\text{and}\quad A_+ y = c$

it is interesting to know if the solutions $$x$$ and $$y$$ to theses systems can be computed approximately by solving the large but finite linear systems

$A_m x^{(m)} = b^{(m)} \quad\text{and}\quad (A_+)_m y^{(m)} = c^{(m)}$

and letting $$m \to \infty$$. This is the main idea of the Finite Section Method (FSM). In order to assure the applicability of the above procedure, one investigates further properties of the operator $$A$$, the sequence $$(A_n)$$ and its one-sided counterparts. In particular, Fredholm Theory, spectral theory and the concept of limit operators play a central role in this investigation [Lind06].

This research project deals with the investigation of the applicability of the FSM to problems surging from aperiodic discrete Schrödinger Operators [GGGU21]. A famous example for theses operators is the so called Fibonacci-Hamiltonian [LiSö18], where the potential $$v$$ is given as

$v(n) := \chi_{[1 - \alpha, 1)}(n \alpha \operatorname{mod} 1)\;, \quad n \in \mathbb{Z}.$

For this particular example, the central objects of investigation are periodic approximations $$(A_m)$$. It is crucial to assure that the spectrum of these approximations eventually avoids the point $$0$$ for larger numbers of $$m$$. The following graph shows approximations of the spectra of the one-sided Fibonacci Hamiltonian on $$\ell^2(\mathbb{N})$$.

## References

[GGGU21]
Gabel, Fabian ; Gallaun, Dennis ; Großmann, Julian ; Ukena, Riko: Finite section method for aperiodic Schrödinger operators.
[Lind06]
Lindner, M.: Infinite Matrices and their Finite Sections: An Introduction to the Limit Operator Method, Frontiers in Mathematics : Birkhäuser Basel, 2006 ? ISBN 9783764377670
[LiSö18]
Lindner, Marko ; Söding, Hagen: Finite sections of the Fibonacci Hamiltonian. In: Operator theory, 2018, S. 381?396