Zeev Rudnick (Tel-Aviv)
Abstract: 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).