New Horizons

You can get very interesting result when plotting different linear combination of eigenfunctions of the Laplacian on a given (convex) shape. Here I use a square plate [0,1] by [0,1] and show the zero iso-value of the some of such linear combination. As time evolves, the linear combination changes. Since the Eigenfunctions are symmetric, I only plot them on the [0, 0.5] interval.  In that case, the eigenfunction are of the form With (n1,n2) such that The associated eigenvalue l= 340*pi^2 is of multiplicity 4 (12,14) (18,4) (14,12) (4,18). Below, same thing with l = 5525 which is of multiplicity 12. Those isocontours are called nodal lines. By the unique continuation property, those nodal lives consist of curves in the interior of the domain (her a square, but the result is valid for any convex shape) which are infinetly differentiable. You may also note, but that is a well known theorem, that when nodal lines interesect a boundary (except may be at corner points...

As usual, I started this morning by browsing arxiv, watching for article that would potentially have the ability to wake me up from my sleepiness. And I found 'Asymptotic rate of quantum ergodicity inchaotic Euclidean billiards'   with the following picture that shocked me. This image results from tracing cumulative plane wave with random orientation, but same wave-number on a square domain and I found strinking how similar to simulation s in cosmology it was. The second image result from simualtion and we see density fluctuation of a simulated universe. The filaments represent families of galaxies, and black zone, the emptiness of space.