Commit a1fe2276 authored by Fabian Nuraddin Alexander Gabel's avatar Fabian Nuraddin Alexander Gabel 💬
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make changes to work with pandoc 2.13

parent 89494792
......@@ -29,12 +29,12 @@ v(n) := \chi_{[1 - \alpha, 1)}(n \alpha \operatorname{mod} 1)\;, \quad n \in \ma
<h2 class="unnumbered" id="references">References</h2>
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-gggu2021" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[GGGU21] </div><div class="csl-right-inline"><span class="smallcaps">Gabel, Fabian</span> ; <span class="smallcaps">Gallaun, Dennis</span> ; <span class="smallcaps">Großmann, Julian</span> ; <span class="smallcaps">Ukena, Riko</span>: <a href="https://arxiv.org/abs/2104.00711">Finite section method for aperiodic Schrödinger operators</a>.</div>
<div class="csl-left-margin">[GGGU21] </div><div class="csl-right-inline"><span class="smallcaps">Gabel, Fabian</span> ; <span class="smallcaps">Gallaun, Dennis</span> ; <span class="smallcaps">Großmann, Julian</span> ; <span class="smallcaps">Ukena, Riko</span>: Finite section method for aperiodic Schrödinger operators.</div>
</div>
<div id="ref-lindner2006infinite" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[Lind06] </div><div class="csl-right-inline"><span class="smallcaps">Lindner, M.</span>: <em><a href="https://books.google.de/books?id=EZxK9rCpUOgC">Infinite Matrices and their Finite Sections: An Introduction to the Limit Operator Method</a></em>, <em>Frontiers in Mathematics</em> : Birkh<span>ä</span>user Basel, 2006 — ISBN 9783764377670</div>
<div class="csl-left-margin">[Lind06] </div><div class="csl-right-inline"><span class="smallcaps">Lindner, M.</span>: <em>Infinite Matrices and their Finite Sections: An Introduction to the Limit Operator Method</em>, <em>Frontiers in Mathematics</em> : Birkh<span>ä</span>user Basel, 2006 — ISBN <a href="https://worldcat.org/isbn/9783764377670">9783764377670</a></div>
</div>
<div id="ref-lindner2018" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[LiSö18] </div><div class="csl-right-inline"><span class="smallcaps">Lindner, Marko</span> ; <span class="smallcaps">Söding, Hagen</span>: <a href="http://hdl.handle.net/11420/3456">Finite sections of the Fibonacci Hamiltonian</a>. In: <em>Operator theory</em>, 2018, S. 381–396</div>
<div class="csl-left-margin">[LiSö18] </div><div class="csl-right-inline"><span class="smallcaps">Lindner, Marko</span> ; <span class="smallcaps">Söding, Hagen</span>: Finite sections of the Fibonacci Hamiltonian. In: <em>Operator theory</em>, 2018, S. 381–396</div>
</div>
</div>
<h1 id="finite-sections-of-aperiodic-schrödinger-operators">Finite Sections of Aperiodic Schrödinger Operators</h1>
<h3 id="working-groups-aa">Working Groups: aa</h3>
<h3 id="collaborators-mat-dgallaun-fgabel-jgrossmann-mlindner-rukena">Collaborators (MAT): dgallaun, fgabel, jgrossmann, mlindner, rukena</h3>
<h2 id="description">Description</h2>
<p>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 <span class="math inline">\(p \in [1,\infty]\)</span>, consider the Schrödinger operator <span class="math inline">\(H \colon \ell^p(\mathbb{Z}) \to \ell^p(\mathbb{Z})\)</span> given by</p>
<p><span class="math display">\[
(H x)_n = x_{n + 1} + x_{n - 1} + v(n) x_nn \in \mathbb{Z},
\]</span>(1)</p>
<p>and its one-sided counterpart <span class="math inline">\(H_+ \colon \ell^p(\mathbb{N}) \to \ell^p(\mathbb{N})\)</span> given by</p>
<p><span class="math display">\[
(H_+ x)_n = x_{n + 1} + x_{n - 1} + v(n) x_n\;,n \in \mathbb{N}, \quad x_0 = 0\;.
\]</span> (2)</p>
<p>Based on Definitions <span class="math inline">\((1)\)</span> and <span class="math inline">\((2)\)</span>, one can associate <span class="math inline">\(H\)</span> and <span class="math inline">\(H_+\)</span> with infinite tridiagonal matrices <span class="math inline">\(A = (a_{ij})_{i,j \in \mathbb{Z}}\)</span> and <span class="math inline">\(A_+ = (a_{i,j})_{i,j \in \mathbb{N}}\)</span>.</p>
<p>Looking at the corresponding infinite linear system of equations</p>
<p><span class="math display">\[
A x = b \quad\text{and}\quad A_+ y = c
\]</span></p>
<p>it is interesting to know if the solutions <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span> to theses systems can be computed approximately by solving the large but finite linear systems</p>
<p><span class="math display">\[
A_m x^{(m)} = b^{(m)} \quad\text{and}\quad (A_+)_m y^{(m)} = c^{(m)}
\]</span></p>
<p>and letting <span class="math inline">\(m \to \infty\)</span>. 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 <span class="math inline">\(A\)</span>, the sequence <span class="math inline">\((A_n)\)</span> and its one-sided counterparts. In particular, Fredholm Theory, spectral theory and the concept of limit operators play a central role in this investigation <span class="citation" data-cites="lindner2006infinite">[Lind06]</span>.</p>
<p>This research project deals with the investigation of the applicability of the FSM to problems surging from aperiodic discrete Schrödinger Operators <span class="citation" data-cites="gggu2021">[GGGU21]</span>. A famous example for theses operators is the so called Fibonacci-Hamiltonian <span class="citation" data-cites="lindner2018">[LiSö18]</span>, where the potential <span class="math inline">\(v\)</span> is given as</p>
<p><span class="math display">\[
v(n) := \chi_{[1 - \alpha, 1)}(n \alpha \operatorname{mod} 1)\;, \quad n \in \mathbb{Z}.
\]</span></p>
<p>For this particular example, the central objects of investigation are periodic approximations <span class="math inline">\((A_m)\)</span>. It is crucial to assure that the spectrum of these approximations eventually avoids the point <span class="math inline">\(0\)</span> for larger numbers of <span class="math inline">\(m\)</span>. The following graph shows approximations of the spectra of the one-sided Fibonacci Hamiltonian on <span class="math inline">\(\ell^2(\mathbb{N})\)</span>.</p>
<p><img src="img/periodicApproxFibHam.png" /></p>
<h2 class="unnumbered" id="references">References</h2>
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
<div id="ref-gggu2021" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[GGGU21] </div><div class="csl-right-inline"><span class="smallcaps">Gabel, Fabian</span> ; <span class="smallcaps">Gallaun, Dennis</span> ; <span class="smallcaps">Großmann, Julian</span> ; <span class="smallcaps">Ukena, Riko</span>: <a href="https://arxiv.org/abs/2104.00711">Finite section method for aperiodic Schrödinger operators</a>.</div>
</div>
<div id="ref-lindner2006infinite" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[Lind06] </div><div class="csl-right-inline"><span class="smallcaps">Lindner, M.</span>: <em><a href="https://books.google.de/books?id=EZxK9rCpUOgC">Infinite Matrices and their Finite Sections: An Introduction to the Limit Operator Method</a></em>, <em>Frontiers in Mathematics</em> : Birkh<span>ä</span>user Basel, 2006 — ISBN 9783764377670</div>
</div>
<div id="ref-lindner2018" class="csl-entry" role="doc-biblioentry">
<div class="csl-left-margin">[LiSö18] </div><div class="csl-right-inline"><span class="smallcaps">Lindner, Marko</span> ; <span class="smallcaps">Söding, Hagen</span>: <a href="http://hdl.handle.net/11420/3456">Finite sections of the Fibonacci Hamiltonian</a>. In: <em>Operator theory</em>, 2018, S. 381–396</div>
</div>
</div>
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