In this talk we address the nodal count (i.e., the number of nodal domains)
for eigenfunctions of Schroedinger operators with Dirichlet boundary
conditions in bounded domains (billiards). The classical Sturm theorem
claims that in dimension one, the nodal and eigenfunction counts coincide: the n-th
eigenfunction has exactly n nodal domains. The Courant Nodal Theorem
claims that in any dimension, the number of nodal domains of the n-th
eigenfunction cannot exceed n. However, it follows from a stronger upper bound by
Pleijel that in dimensions higher than 1 the equality can hold for only
finitely many eigenfunctions. Thus, in most cases a ``nodal deficiency''
arises. Moreover, examples are known of eigenfunctions with an arbitrarily
large index n that have just two nodal domains. One can say that the
nature of the nodal deficiency had not been understood.
We show that, under some genericity conditions, the answer can be given in
terms of a functional on an infinite dimensional variety of partitions of
the billiard, whose critical points correspond exactly to the nodal
partitions and Morse indices coincide with the nodal deficiencies.
This is joint work with G. Berkolaiko and U. Smilansky.
