Global Harmonic Analysis

Northwestern University, Evanston IL

Investigators

Abstract

The fundamental equations of quantum mechanics contain a parameter h known as Planck's constant; the observed value of h is understood to be one of the fundamental constants of our universe. Relationships between quantum-mechanical systems and their classical counterparts can be explored by investigating how the quantum-mechanical equations behave as the value of the parameter h tends to zero in some sense. Pursuing this idea rigorously, however, involves subtleties. Because h is not a dimensionless quantity, the "smallness" of values of the parameter must be expressed in terms of other relevant quantities. This research project aims to deepen understanding of relationships between quantum mechanics and classical mechanics through rigorous mathematical analysis centered on the limit when the Planck constant h tends to zero. The questions under study are of great interest both to mathematicians and to physicists. This research project primarily concerns stationary states of quantum systems. 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 goals of the project are (1) to analyze Lp norms of eigenfunctions and (2) to analyze nodal sets and critical point sets. Making use of recent results relating geometric control and restrictions of eigenfunctions to hypersurfaces, the investigator aims to study nodal problems in non-ergodic situations and possible relations to the fractal uncertainty principle. The project also includes exploration of several other topics, including the numbers of nodal domains of eigenfunctions on manifolds of higher dimension; Lp norms of eigenfunctions and microlocal Kakeya-Nikodym norms and their relations to analytic continuation to Grauert tubes; and the behavior of density of states of spectral projections for various Hamiltonians across natural interfaces such as energy surfaces. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.