Given a closed manifold, the Laplace operator is know to possess a discrete spectrum of eigenvalues converging to infinity. We are interested in properties of the corresponding eigenfunctions as the eigenvalue becomes large
(i.e. the high-energy limit). From a physical point of view, the eigenfunctions represent stationary states of a free quantum particle—when properly normalized, they may be interpreted as the probability density of a particle in the manifold.
Various questions about the geometry of Laplace-eigenfunctions have been studied thoroughly—for example, distribution and measure of the vanishing (nodal) set; localization; shape and inner radius of nodal domains, etc.
We present some recent results along these lines—this also includes joint work with M. Mukherjee.