Global Harmonic Analysis

Northwestern University, Evanston IL

Investigators

Abstract

The PI shall prove rigorous results about the relations between classical and quantum mechanics. Schrodinger introduced quantum mechanics in 1927 to solve the puzzle of how an electron can be moving and stationary at the same time. His answer was that a quantum particle with a definite energy can only exist in a stationary state, the physics word for an eigenfunction of the Schrodinger operator governing the physics. The problem is that eigenfunctions are very hard to visualize and understand. Global Harmonic Analysis is about making rigorous relations between these eigenfunctions and the classical mechanics underlying the physical system. The projects in the proposal address two opposite kinds of questions: (i) how large can the eigenfunction be, and how are the points where it is large distributed? These are the points where it is most probable to find the particle. (ii) How are the points where the eigenfunction is zero distributed? These are the points where it is least likely to find the particle. Global Harmonic Analysis is the use of the long-time dynamics of the geodesic flow of a Riemannian manifold to understand the high-frequency asymptotics of eigenfunctions. Two of the most important problems are (1) to analyze Lp norms of eigenfunctions and (2) to analyze nodal sets and critical point sets. For large p, Sogge and the PI have shown that the universal Lp bounds are only obtained if there exists a self-focal point x where a positive measure of geodesics from x loop back to x. In recent work, The PI showed that if the metric is real analytic, then all geodesics loop back and the first return map preserves a finite measure on the unit tangent space at x in the class of Lebesgue measure. In two dimensions, it follows that all geodesics through x are smoothly closed. One project is to generalize the last result to higher dimensions and smooth metrics. It is also very interesting to study low Lp norms and we are working with the Kakeya-Nikodym maximal function to do that. Regarding nodal and critical point sets, J. Jung and the PI have recently shown that the number of nodal domains of Neumann or Dirichlet eigenfunctions on non-positively curved surfaces with non-empty concave boundary must tend to infinity with the eigenvalue. It was also showed that the eigenfunctions have many critical points. In future work, the PI shall prove a more quantitative result and to study the higher dimensional case. The PI is also using holomorphic extensions of eigenfunction in the real analytic case to gain more control over nodal and critical point sets.