Merge branch 'aperiodSchr'

topics/aperiodSchr.md

+# Finite Sections of Aperiodic Schrödinger Operators
+
+### Working Groups: aa
+
+### Collaborators (MAT): dgallaun, fgabel, jgrossmann, mlindner, rukena
+
+## 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 [1].
+
+This research project deals with the investigation of the applicability of the FSM to problems surging from aperiodic discrete Schrödinger Operators [3]. 
+A famous example for theses operators is the so called Fibonacci-Hamiltonian [2], 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 $\ell^2(\mathbb{N})$.
+
+![](/img/periodicApproxFibHam.png)
+
+## References
+
+ [1] M. Lindner. Infinite Matrices and their Finite Sections: An Introduction to the Limit
+Operator Method, Basel: Birkhäuser, 2006.
+
+ [2] M. Lindner, H. Söding. "Finite Sections of the Fibonacci Hamiltonian," in The Diversity and Beauty of Applied Operator Theory, edited by A. Böttcher, D. Potts, P. Stollmann and D. Wenzel. Cham: Springer International Publishing, (2018):381-396. 
+
+ [3] F. Gabel, D. Gallaun, J. Großmann, R. Ukena. The Finite Section Method for Aperiodic Schrödinger Operators. [arXiv:2104.00711](https://arxiv.org/abs/2104.00711)
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