### remove test_topics

parent a9867ba6
Pipeline #116717 passed with stages
in 33 seconds
 --- bibliography: bib/aperiodSchr.bib csl : csl/din-1505-2-alphanumeric.csl --- # 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 [@lindner2006infinite]. This research project deals with the investigation of the applicability of the FSM to problems surging from aperiodic discrete Schrödinger Operators [@gggu2021]. A famous example for theses operators is the so called Fibonacci-Hamiltonian [@lindner2018], 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
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!