Nodal Intersections

We study the fine structure of nodal lines for eigenfunctions of the Laplacian on a a surface by examining the number of intersection of the nodal lines with a fixed reference curve. It is expected that in many cases the number of these intersections is bounded above by the wave number $k$ (the square root of the eigenvalue). Very little is known concerning lower bounds. For the flat torus, we prove the expected upper bound of $k$ and give a lower bound of almost the same quality. To do so, we connect this problem to bounds on the Lp norms of the restriction of the eigenfunctions to the curve, and to a problem in Number Theory.

(joint work with Jean Bourgain).