Computing eigenvalues for quantum maps

This page contains additional material to the paper Numerical aspects of eigenvalue and eigenfunction computations for chaotic quantum systems. To use the following examples you will need Python and Numeric, see below for further details.

Quantum map programm

We will use a perturbed cat map as an example. For the following you will need to download

The core of these two modules is shown on this page. As a first test do (for N=101 and kappa=0.3)

    python pert_cat.py 101 0.3

It will output the (complex) eigenvalues as a sequence of (x,y) pairs.
(If there is an error, see installation.)
As a test, whether these all lie on the unit circle the fourth column is the absolute value of the eigenvectors.
To generate a plot of the resulting data you could first use

    python pert_cat.py 101 0.3 > pcat_101_0.3.dat

which redirects the output of the programm to the file pcat_101_0.3.dat.
To plot the resulting file use your favourite plotting programm, e.g. in gnuplot just do

    plot "pcat_101_0.3.dat" using 1:2 with points

Level spacing distribution

Now we would like to compute the level spacing distribution. To do this let us use an interactive Python session in which we do (usually copying-and-pasting the full code will lead to problems, so either copy line-by-line, or download compute_spacings.py and do python compute_spacings.py)

from AnalyseData import histogram,store_histogram
import Numeric
from math import pi
import pert_cat

N=53
#N=501
kappa=0.3 
(evals,phases)=pert_cat.compute_evals_pcat(N,kappa);
# sort and unfold phases
s_phases=Numeric.sort(phases)*N/(2.0*pi)

# determine Level spacing
# (by computing the difference of the shifted eigenphases)
spacings=s_phases[1:]-s_phases[0:N]
 
(x_histogram,y_histogram)=histogram(spacings,0.0,10.0,100)
store_histogram(x_histogram,y_histogram,"histogram.dat")

Then use your favourite plotting programm to plot the spacing histogram. For gnuplot you could do

    goe_approx(x)=pi/2.0*x*exp(-pi/4*x*x)
    gue_approx(x)=32/pi/pi*x*x*exp(-4/pi*x*x)
    plot "histogram.dat" w l,goe_approx(x),gue_approx(x),exp(-x)

Some pointers on Python

Requirement for the above example is (installation is really easy!)

For interactive work I highly recommend to use

IPython

For example you can then easily use the above gnuplot commands within an ipython session!

Introductions, overviews, programming tutorials, ... and much more can be found in the Documentation section on the Python Homepage.

Finally, you might give Scipy a try (which is under heavy development at the moment and will be a good replacement of Matlab or similar tools):

SciPy

Concluding remark

If you have any comments/suggestions/questions etc. don't hesitate to contact me.

Acknowledgements

First I would like to thank Fernando Perez for several useful suggestions and speed improvements on the original version of the code. Comments by Silke Fürstberger, Rainer Glaser and Grischa Haag improved the usability of these pages.