#eigenfunctions

a density plot of an eigenfunction of the Laplacian on the upper half-plane with the hyperbolic (constant negative-curvature) metric, obeying certain symmetries which make it a so-called arithmetic surface. Black shows large values of the square of the mode value, white zero. Its fundamental domain is the strip shown, bounded by an arc at one end and unbounded on the other. For aesthetic reasons the y-axis has been chosen horizontal; in Cartesian coordinates the area element is dxdy/y2. These eigenfunctions are known as Maass forms. They have importance in number theory, as well as providing a rare example where QUE has been (recently) proven.

An introduction can be found in the Baltimore review article of Peter Sarnak, here.

Below is a set of 12 consecutive such eigenfunctions, with mode number n=5601, 5602,..., 5612, shown in the more usual orientation (y is vertical, and the x-axis is visible as the line segment at the bottom), and with the density now with the correct metric (intensity proportional to square of mode value times 1/y2). The modes were evaluated by Alex Barnett using code, and Fourier coefficients, provided by Holger Then.

Yes?

nostalgebraist replied to your post “does anybody on here wanna think about eigenfunctions”

what about them?

I know sort of scattered things about eigenfunctions, like some stuff about quantum ergodicity, a little bit about lower bounds on Hausdorff measure of nodal sets (mostly from reading an amazingly easy to understand paper by Sogge and Zelditch) and other stuff I’ve picked up from casually reading some notes by Zelditch. But I’m just now starting to read a little more carefully and systematically, reading Sogge’s book, and uh it’s a great book. Maybe I’ve just been looking at Hörmander too much, but so far everything in here is really straightforward and easy to understand.