This repository is push mirrored to https://github.com/eltenedor/simulation_aperiodSchr

Spectral Analysis Code for Aperiodic Schrödinger Operators

Simulation code and supplementary material for the article "Finite Section Method for Aperiodic Schrödinger Operators".

arXiv-Preprint: https://arxiv.org/abs/2104.00711

Lower Norm

List of Octave/MATLAB files:

Filename	Info
`LowerNormsTest.m`	a script that runs tests for `checkLowerNorms`
`bandsPeriodicScaledTwoSided.m`	calculating the spectral bands scaled periodic approximations
`checkLowerNorms.m`	implementation of Algorithm 1, checks if the condition (32) in Theorem 5.8 is satisfied for fixed D
`constructAllWords.m`	returns all possible subwords of v_{\alpha,\theta} of length k
`constructWord.m`	returns s_n, see equation (9)
`copyWord.m`	auxiliary function
`growingTruncations.m`	used for the generation of Figure 3
`plotPeriodicApproximation.m`	script used for the generation of Figure 2
`pointSpecPeriodicScaledOneSided.m`	calculating the point spectra of scaled periodic approximations
`smallest_sv.m`	computes the smallest singular value of a rectangular matrix with diagonal v and sub- and superdiagonal that are constant 1
`tridiagDet.m`	Recursion to calculate determiant of tridiagonal matrix
`unionPointSpecPeriodicScaledOneSided.m`	calculating the union of point spectra of scaled periodic approximations
`createTex....m`, `postproc_....m`	auxiliary routines uses for TikZ output

Example 5.9

This Code was tested with GNU Octave, version 6.4.0

To reconstruct Example 5.9, in Octave/MATLAB run

[isApplicable, minLowerNorm, delta, words, lowerNorms] = checkLowerNorms(1, 100, ones(11,1));

and check that

octave:2> isApplicable
isApplicable = 1

Figure 2

This Code was tested with GNU Octave, version 6.4.0

In order to reproduce Figure 2, in Octave/MATLAB run

plotPeriodicApproximation.m

and check that the script finishes

...
100% complete

The content below is relevant for version 2 of the preprint

https://arxiv.org/abs/2104.00711v2

as the content is no longer included in subsequent versions of the preprint.

Point and Band Spectrum

The directory /band_and_point_spec contains the SageMath Jupyter Notebooks and Octave scripts that were used for the proof of Theorem 1.1 in [GGGLU21] and several figures.

Proof of Theorem 1.1

The data for the proof was generated using the notebook potential_data.ipynb. The JSON file pots.json contains all the data needed for the proof of Theorem 1.2. All results in pots.json only use symbolic calculations carried out with SageMath. For further documentation, consult the documentation in the notebook potential_data.ipynb.

Figures 2, 3, 4

The data for the figures resides in the subdirectory /archive.

Reproducibility

In order to reproduce

the band spectral data, run the jupyter notebook bands.ipynb.
the point spectral data, run the jupyter notebook point_spec.ipynb.

Both routines will generate .csv data in the /data folder that is further processed by Octave scripts to generate TikZ code.

In order to create the tikz code needed for the plots, run the postproc_KX.m Octave scripts, where X is either 4, 5 or 9 depending on the length of the example potential that you are interested in. All the newly produced data will reside in the subdirectory /data together with auxiliary .tex files. In order to reproduce one of the figures, run

pdflatex KX-standalone.tex

where X is either 4, 5 or 9 depending on the length of the example potential that you are interested in.

Modifications in the case K=9

In order to account for the multiplicities of the zeros for the scalings l=0 and l=2, the file output of the K9 cells in bands.ipynb needs to be changed before the postprocessing can start.

In the output file bands_K9.csv, change the line for l = 0 to

0.000000000000000,9,-2.00000000000000,-1.87938524157182,-1.87938524157182,-1.53208888623796,-1.53208888623796,-1.00000000000000,-1.00000000000000,-0.347296355333861,-0.347296355333861,0.347296355333861,0.347296355333861,1.00000000000000,1.00000000000000,1.53208888623796,1.53208888623796,1.87938524157182,1.87938524157182,2.00000000000000

and the one for l=2 to

2.00000000000000,9,-1.12269503546099,-1.11122114140128,-0.801937735804838,-0.762512708829868,-0.341677503250975,-0.288537854368133,0.554958132087371,0.614111698460124,1.56339706474929,1.67439549622911,2.24697960371747,2.48317863579858,2.66734798160450,3.00000000000000,3.00000000000000,3.18291284403670,3.71593700828199,3.72536349954030

See also the output of the corresponding cell in bands.ipynb.

Authors

Name	Mail
Fabian Gabel	fabian.gabel@tuhh.de
Dennis Gallaun	dennis.gallaun@tuhh.de
Julian Großmann	julian.grossmann@jp-g.de
Marko Lindner	lindner@tuhh.de
Riko Ukena	riko.ukena@tuhh.de

References

The above code was tested using Octave 6.4.0

  John W. Eaton, David Bateman, Søren Hauberg, Rik Wehbring (2019).
  GNU Octave version 6.4.0 manual: a high-level interactive language for numerical computations.
  URL https://www.gnu.org/software/octave/doc/v6.2.0/

and SageMath version 9.2

  William A. Stein et al. (2020) 
  Sage Mathematics Software (Version 9.2), The Sage Development Team
  URL http://www.sagemath.org.

Parts of this code also reproduces results in the preprint [GGGLU21]

Fabian Gabel, Dennis Gallaun, Julian Großmann, Marko Lindner, Riko Ukena (2021). 
Finite section method for aperiodic Schrödinger operators. 
arXiv preprint arXiv:2104.00711.

fabiangabel/simulation_aperiodSchr