Refactor code into smaller pieces to avoid repetitive patterns

.gitignore

 build/*
+*/build
 local/*
 public/*
 *.html
+*.log
+

.gitlab-ci.yml

@@ -2,8 +2,28 @@
 # Full project: https://gitlab.com/pages/plain-html
 #
 
+stages:
+  - preprocessor-test
+  - preprocessor-run
+  - pandoc-run
+  - deploy
+
+testing:
+  stage: preprocessor-test
+  allow_failure: false
+  image: 
+    name: alpine:latest
+  script:
+    - apk add --no-cache --upgrade bash
+    - RESEARCH_ROOT=`pwd`; export RESEARCH_ROOT;
+    # run shell script with testss
+    - $RESEARCH_ROOT/bin/run-tests.sh
+  artifacts:
+    paths:
+      - testing/*.log
+
 preprocessing:
-  stage: build
+  stage: preprocessor-run
   allow_failure: false
   image: 
     name: alpine:latest
@@ -11,14 +31,13 @@ preprocessing:
     - apk add --no-cache --upgrade bash
     - RESEARCH_ROOT=`pwd`; export RESEARCH_ROOT;
     # preprocess .MD files
-    - $RESEARCH_ROOT/bin/preprocIndexMd.sh
-    - $RESEARCH_ROOT/bin/preprocMd.sh
+    - $RESEARCH_ROOT/bin/preproc_topics.sh
   artifacts:
     paths:
       - build
 
 pandoc: 
-  stage: test
+  stage: pandoc-run
   dependencies: [preprocessing]
   image:      
     name: pandoc/latex:latest

bin/buildWebpage.sh

@@ -9,8 +9,7 @@ rm -rf  $BUILD_DIR
 rm -rf $DEPLOY_DIR/{.public,public}
 
 #build Index
-$BIN_DIR/preprocIndexMd.sh
-$BIN_DIR/preprocMd.sh
+$BIN_DIR/preproc_topics.sh
 $BIN_DIR/buildPandoc.sh
 
 #make public directory and copy files

bin/extract_title.sh

+#!/bin/bash
+# extract title of research topic
+
+filename=$(basename -- "$1")
+
+echo "## `head -n 1 $1 | sed -e "s;#\s*;\[;g" -e "s;\(.*\);\1\](${filename%.md}.html);g"`"
\ No newline at end of file

bin/preproc_staff.sh

+#!/bin/bash
+# preproc_staff.sh
+# Preprocessor for staff keys in .md topic file
+# Replaces key fgabel by "Fabian Gabel, M.Sc." according to Webpage content
+# If Build_dir is passed, titles are appended to corresponding files
+# Note: mat.tuhh.de uses WINDOWS-1252 character encoding
+
+# processing command line arguments
+if [ $# -lt 2 ]
+then
+    RESEARCH_BUILD=`pwd`"/build"
+    echo -e "No building directory was specified."
+else
+    RESEARCH_BUILD="$2"
+fi
+mkdir -p $RESEARCH_BUILD
+echo -e "Building into directory $RESEARCH_BUILD ..."
+
+filename=$(basename -- "$1")
+baseurl='https://www.mat.tuhh.de'
+#make output copy
+echo "Preprocessing $filename..."
+if [ -f "$RESEARCH_BUILD/$filename" ]
+then
+    echo "File exists, performing preprocessing in place"
+else
+    cp $1 $RESEARCH_BUILD/$filename
+fi
+
+#extract title of topic
+echo "Extracting title of $1..."
+title=`./extract_title.sh $RESEARCH_BUILD/$filename`
+echo "Title of Topic: $title"
+
+# start preprocessing 
+# -- leave original untouched, only work with copy in $RESEARCH_BUILD
+echo "Preprocessing collaborators in file $RESEARCH_BUILD/$filename ..."
+namelist=`grep -h -i -m 1 -r "###\s*Collaborators (MAT):" $RESEARCH_BUILD/$filename | sed -e 's/^###\s*Collaborators (MAT):\s*//I' -e 's/\s*,\s*/\n/g'  | sort -u`
+echo "Found the following keys: " $namelist 
+echo "Replacing collaborator keys ..."
+for name in $namelist
+do
+    # pipeline to get full name of staff-member from mat-homepage
+    # -> wget the staff homepage of $name
+    # -> grep the line with the <h1>-tag, something like <h1>Fabian Gabel, M. Sc.</h1><div class='staffIntro'><p><img src='/home/fgabel/images/portrait.png' title='Foto von Fabian Gabel, M. Sc.' class='staffPicture'></p><div class='staffContact'>
+    # -> strip the string such that only the portion between <h1></h1> remains
+    # -> remove leading spaces
+    wget -qO- $baseurl/home/$name/?homepage_id=$name > page.html
+    iconv -f WINDOWS-1252 -t UTF-8 ./page.html > ./utf.html
+    fullname=`grep h1 ./utf.html |  sed -e "s/<h1>\s*\(.*\)<\/h1>.*$/\1/g" | sed -e 's/^[ \t]*//'`
+    rm -rf {utf,page}.html
+
+    echo "Found collaborator $fullname" 
+
+    # replace name in .md file
+    sed -i "s;$name;\[$fullname\]($name.html);g" $RESEARCH_BUILD/$filename 
+    #sed -i "s;$name;\[$fullname\]($baseurl/home/$name);g" $RESEARCH_BUILD/$filename #uncomment this line for linking the mat.tuhh.de webpage
+
+    # delete external collaborators if left empty
+    sed -i -e '/###\s*Collaborators (External):\s*$/ d' $RESEARCH_BUILD/$filename
+
+    if [ $# -gt 1 ]
+    then
+        # append research to staffile
+        echo -e "\n$title\n" >> $RESEARCH_BUILD/$name.md
+    fi
+done
+
+# adpat img path (prefix a dot)
+sed -i "s;\](/img/;\](./img/;g" $RESEARCH_BUILD/$filename 
\ No newline at end of file

bin/preproc_topics.sh

+#!/bin/bash
+
+RESEARCH_BUILD="$RESEARCH_ROOT/build"
+rm -rf $RESEARCH_ROOT/build
+
+mkdir -p $RESEARCH_BUILD
+baseurl='https://www.mat.tuhh.de'
+
+#build index.html
+#replace working group
+cp $RESEARCH_ROOT/index.md $RESEARCH_BUILD/index.md
+$RESEARCH_ROOT/bin/preproc_wg.sh $RESEARCH_BUILD/index.md $RESEARCH_BUILD >> $RESEARCH_BUILD/build.log 2>&1
+
+#replace staff names
+$RESEARCH_ROOT/bin/preproc_staff.sh $RESEARCH_BUILD/index.md $RESEARCH_BUILD >> $RESEARCH_BUILD/build.log 2>&1
+
+#finalize file
+sed -i "s;\](/img/;\](./img/;g" $RESEARCH_BUILD/index.md
+
+
+#build rest of the topics
+for f in $RESEARCH_ROOT/topics/*.md
+do
+    echo $f
+    filename=$(basename -- "$f")
+    #make output copy
+    echo "Preprocessing $filename..."
+    cp $f $RESEARCH_BUILD/$filename
+
+    #extract title of topic
+    title=`./extract_title.sh $RESEARCH_BUILD/$filename`
+    echo "Preprocessing topic $title"
+
+    # append research topic to index
+    echo -e "\n$title\n" >>  $RESEARCH_BUILD/index.md
+
+    #replace working group
+    $RESEARCH_ROOT/bin/preproc_wg.sh $RESEARCH_BUILD/$filename $RESEARCH_BUILD >> $RESEARCH_BUILD/build.log 2>&1
+
+    #replace staff names
+    $RESEARCH_ROOT/bin/preproc_staff.sh $RESEARCH_BUILD/$filename $RESEARCH_BUILD >> $RESEARCH_BUILD/build.log 2>&1
+
+    #finalize file
+    sed -i "s;\](/img/;\](./img/;g" $RESEARCH_BUILD/$filename
+done
\ No newline at end of file

bin/preproc_wg.sh

+#!/bin/bash
+# preproc_staff.sh
+# Preprocessor for staff keys in .md topic file
+# Replaces key fgabel by "Fabian Gabel, M.Sc." according to Webpage content
+# If Build_dir is passed, titles are appended to corresponding files
+# Note: mat.tuhh.de uses WINDOWS-1252 character encoding
+
+# processing command line arguments
+if [ $# -lt 2 ]
+then
+    RESEARCH_BUILD=`pwd`"/build"
+    echo -e "No building directory was specified."
+else
+    RESEARCH_BUILD="$2"
+fi
+mkdir -p $RESEARCH_BUILD
+echo -e "Building into directory $RESEARCH_BUILD ..."
+
+filename=$(basename -- "$1")
+baseurl='https://www.mat.tuhh.de'
+#make output copy
+echo "Preprocessing $filename..."
+if [ -f "$RESEARCH_BUILD/$filename" ]
+then
+    echo "File exists, performing preprocessing in place"
+else
+    cp $1 $RESEARCH_BUILD/$filename
+fi
+
+#extract title of topic
+echo "Extracting title of $1..."
+title=`./extract_title.sh $RESEARCH_BUILD/$filename`
+echo "Title of Topic: $title"
+
+# start preprocessing 
+# -- leave original untouched, only work with copy in $RESEARCH_BUILD
+# replace working group
+echo "Replacing working group keys..."
+namelist=`grep -h -i -m 1 -r "###\s*Working Groups:" $RESEARCH_BUILD/$filename | sed -e 's/^###\s*Working Groups:\s*//I' -e 's/\s*,\s*/\n/g'  | sort -u`
+echo "Found the following keys: " $namelist
+for wg in $namelist
+do
+    #extract full name from tuhh-webpage
+    fullname=`wget -qO- $baseurl/forschung/$wg | grep h1 | sed -e "s;<h1>\s*\(.*\)</h1>.*$;\1;g" | sed -e 's;^[ \t]*;;'`
+
+    echo "Found working group $fullname"
+
+    # append research to wg-file
+    if [ $# -gt 1 ]
+    then
+    echo -e "\n$title\n" >> $RESEARCH_BUILD/$wg.md
+    fi
+
+    sed -i "s;### Working Groups:\(.*\)$wg\(.*\);### Working Groups:\1\[$fullname\]($wg.html)\2;g" $RESEARCH_BUILD/$filename  
+    #sed -i "s;### Working Groups:\(.*\)$wg\(.*\);### Working Groups:\1\[$fullname\]($baseurl/forschung/$wg)\2;g" $RESEARCH_BUILD/$filename  #uncomment this line for linking the mat.tuhh.de webpage
+done
\ No newline at end of file

bin/run-tests.sh

+#!/bin/bash
+
+# set environment variable for testing dir
+RESEARCH_TESTING=$RESEARCH_ROOT/testing
+rm -rf $RESEARCH_TESTING/*.log
+
+#BEGIN Test 1: Preprocessor for Staff 
+rm -rf $RESEARCH_TESTING/build
+echo "Test 1: Preprocessor for staff keys"
+$RESEARCH_ROOT/bin/preproc_staff.sh $RESEARCH_TESTING/aperiodSchr.md $RESEARCH_TESTING/build >> $RESEARCH_TESTING/testing.log 2>&1
+DIFF=$(diff $RESEARCH_TESTING/build/aperiodSchr.md $RESEARCH_TESTING/aperiodSchr-preproc.md) 
+if [ "$DIFF" != "" ] 
+then
+    echo "Test 1: failed"
+    exit 1
+else
+    echo "Test 1: passed."
+fi
+#END Test 1
+
+#BEGIN Test 2: Preprocessor for Working Group
+rm -rf $RESEARCH_TESTING/build
+echo "Test 2: Preprocessor for staff keys"
+$RESEARCH_ROOT/bin/preproc_wg.sh $RESEARCH_TESTING/aperiodSchr.md $RESEARCH_TESTING/build >> $RESEARCH_TESTING/testing.log 2>&1
+DIFF=$(diff $RESEARCH_TESTING/build/aperiodSchr.md $RESEARCH_TESTING/aperiodSchr-preproc-wg.md) 
+if [ "$DIFF" != "" ] 
+then
+    echo "Test 2: failed"
+    exit 1
+else
+    echo "Test 2: passed."
+fi
+#END Test 2
+
+#BEGIN Test 3: Preprocessor Staff then Working Group
+rm -rf $RESEARCH_TESTING/build
+echo "Test 3: Preprocessor for staff and working group keys"
+$RESEARCH_ROOT/bin/preproc_staff.sh $RESEARCH_TESTING/aperiodSchr.md $RESEARCH_TESTING/build >> $RESEARCH_TESTING/testing.log 2>&1
+$RESEARCH_ROOT/bin/preproc_wg.sh $RESEARCH_TESTING/aperiodSchr.md $RESEARCH_TESTING/build >> $RESEARCH_TESTING/testing.log 2>&1
+DIFF=$(diff $RESEARCH_TESTING/build/aperiodSchr.md $RESEARCH_TESTING/aperiodSchr-preproc-both.md) 
+if [ "$DIFF" != "" ] 
+then
+    echo "Test 3: failed"
+    exit 1
+else
+    echo "Test 3: passed."
+fi
+#END Test 3
+
+#BEGIN Test 4: Preprocessor Working Group then Staff (opposite order as Test 3)
+rm -rf $RESEARCH_TESTING/build
+echo "Test 4: Preprocessor for working group and staff keys"
+$RESEARCH_ROOT/bin/preproc_wg.sh $RESEARCH_TESTING/aperiodSchr.md $RESEARCH_TESTING/build >> $RESEARCH_TESTING/testing.log 2>&1
+$RESEARCH_ROOT/bin/preproc_staff.sh $RESEARCH_TESTING/aperiodSchr.md $RESEARCH_TESTING/build >> $RESEARCH_TESTING/testing.log 2>&1
+DIFF=$(diff $RESEARCH_TESTING/build/aperiodSchr.md $RESEARCH_TESTING/aperiodSchr-preproc-both.md) 
+if [ "$DIFF" != "" ] 
+then
+    echo "Test 4: failed"
+    exit 1
+else
+    echo "Test 4: passed."
+fi
+#END Test 4
+
+#this should be the last line (errors exit earlier)
+exit 0
\ No newline at end of file

testing/aperiodSchr-preproc-both.md

+# Finite Sections of Aperiodic Schrödinger Operators
+
+### Working Groups: [Lehrstuhl Angewandte Analysis](aa.html)
+
+### Collaborators (MAT): [Dennis Gallaun, M. Sc.](dgallaun.html), [Fabian Gabel, M. Sc.](fgabel.html), [Dr. Julian Großmann](jgrossmann.html), [Prof. Dr. Marko Lindner](mlindner.html), [Riko Ukena, M. Sc.](rukena.html)
+
+
+## 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)
\ No newline at end of file

testing/aperiodSchr-preproc-wg.md

+# Finite Sections of Aperiodic Schrödinger Operators
+
+### Working Groups: [Lehrstuhl Angewandte Analysis](aa.html)
+
+### Collaborators (MAT): dgallaun, fgabel, jgrossmann, mlindner, rukena
+
+### Collaborators (External): 
+
+## 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)
\ No newline at end of file

testing/aperiodSchr-preproc.md

+# Finite Sections of Aperiodic Schrödinger Operators
+
+### Working Groups: aa
+
+### Collaborators (MAT): [Dennis Gallaun, M. Sc.](dgallaun.html), [Fabian Gabel, M. Sc.](fgabel.html), [Dr. Julian Großmann](jgrossmann.html), [Prof. Dr. Marko Lindner](mlindner.html), [Riko Ukena, M. Sc.](rukena.html)
+
+
+## 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)
\ No newline at end of file

testing/aperiodSchr.md

+# Finite Sections of Aperiodic Schrödinger Operators
+
+### Working Groups: aa
+
+### Collaborators (MAT): dgallaun, fgabel, jgrossmann, mlindner, rukena
+
+### Collaborators (External): 
+
+## 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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