Volume 9, Issue 3
Tricks of the Trade
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T R O T T ' S C O R N E R
The Weyl Law
Finally, we calculate some more eigenvalues. Because the Arnoldi method calculates only some of the eigenvalues, we will call the function drumEigensystem repeatedly to calculate the first 1000 eigenvalues. To keep the memory usage low, we will calculate 24 eigenvalues each time. By aligning the newly calculated eigenvalues with the last set, we are sure not to miss an eigenvalue.
The sum of the first 1000 eigenvectors squared yields the following castle-like graphic. We see a smooth constant background density with relatively small oscillations superimposed.
One of the holy grails of is counting the number of eigenvalues that are less than . According to the classical Weyl law, the number of eigenvalues up to the eigenvalue is given by the asymptotic formula  for smooth boundaries. Here is the area of the drum and the length of its boundary. We see a good overall agreement between the theoretical curve and the calculated cumulative eigenvalue count. At --where the eigenfunctions start to develop nontrivial lateral structure--we see a clear spectral gap and a change in the slope of the calculated . For small , the linear contribution from the length of the circumference dominates the quadratic area contribution.
While for the calculated seems to be linear overall, analyzing the detailed behavior of the terms that contribute to would lead us outside of the realm of this column. The formula provided earlier was for smooth boundaries. For fractal-like boundaries, we have a correction to the volume term of the form instead , where is the fractal dimension of the boundary. And, indeed, a quadratic function fits the corrections slightly better than a linear one.
In conclusion, we leave it to the interested reader to calculate the distribution of the eigenvalue spacings, the autocorrelation functions of the eigenfunctions, and the localization lengths, and to compare them with the properties of a quantum graph corresponding to the drum spine, to add a magnetic field perpendicular to the drum, and so on; or, to go in a more artistic instead of scientific direction and "play" the drum using Play on the eigenvalues.
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