What does a typical (generalized) eigenfunction of the laplacian in the unbounded domain R², ie 2D free space, look like? Say we choose the eigenvalue 1, then the complex plane wave e^in.x, with n is a unit direction vector, is an eigenfunction, but is not typical. A typical sample from the eigenspace with eigenvalue 1 can be constructed by a superposition of such waves traveling in all directions, with random complex phases. In other words, it is the 2D Fourier transform of Gaussian white noise restricted to the unit circle S¹. Berry first proposed this as a model for quantum eigenfunctions of systems whose classical dynamics is chaotic, in the '70s.

Here are some pictures of such typical generalized eigenfunctions, however chosen real-valued, with mean-square value 1, computed with the code linked below. Click on each to magnify:

	color plot	squared	extreme value set (|f|>2.5)
25 wavelengths box
200 wavelengths box

Here is Matlab code to produce the above pictures: rpw2d.tar.

The code is spectrally accurate (with default parameter e=5 it has absolute errors of order 3e-4), and uses the nonuniform FFT by resampling onto a uniform square grid. It is a tar file; unpack then run rpw2d.m. Set n to be 4 times the side length of the square evaluation grid you desire, and set ppw to be the desired number of grid points per wavelength. With n=4096 it takes of order 10 sec on my laptop for all computations, using about 400MB RAM. Time scales like n^2 log n and memory use like n^2; both these are optimal. Enjoy!

Please get in touch if you have ideas how to address the statistical problem of quantifying the `stringiness' visible to the eye in the above.

Some references / ideas:

• ``Properties of random superpositions of plane waves'', P. O'Connor, J. Gehlen, E. J. Heller, Phys. Rev. Lett. 58, 1296-99 (1987). This paper was how I first learned about this, showing that when the wavenumbers are allowed to vary in magnitude, the stringiness disappears.
• Terence Tao's notes on Tomas-Stein theorems on restrictions of the Fourier transform to the sphere. This seems not to be relevant since the functions do not live in L^p spaces.

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