Nodal Sets and Correlations near the Caustic for Fixed Energy Oscillator Eigenfunctions

TYPETheor./Math. Physics Seminar
Speaker:Dr. Boris Hanin
Location:Lewiner Seminar Room (412)
Abstract:This talk concerns the semiclassical behavior of nodal sets and more generally correlations of fixed energy eigenfunctions of the isotropic harmonic oscillator on R^d near the caustic (the boundary between the classically allowed and forbidden regions).
In the allowed region, eigenfunctions rapidly oscillate and behave in many ways like random superpositions of plane waves. In the forbidden region, however, they are exponentially damped. It is therefore unclear to what extent their nodal set persists to infinity and how much they can concentrate near the caustic. When d=1, for example, it is a classical fact that Hermite functions have all their zeros in the allowed region and that Hermite functions have a bump near the caustic. As I will explain, the situation is more complicated in higher dimensions in part because the dimension of the space of fixed energy Hermite functions grows like h^{-d+1}. Joint work with S. Zelditch and P. Zhou