Global harmonic analysis and quantum dynamics

Northwestern University, Evanston IL

Investigators

Abstract

Abstract Award: DMS-1206527 Principal Investigator: Steve Zelditch This projects in this proposal concern global harmonic analysis and asymptotic geometry, with emphases on eigenvalues and eigenfunctions of the Laplacian on Riemannian manifolds, and on Bergman kernels and Bergman metrics in complex geometry. In the Riemannian setting, the main problems concern eigenvalues and eigenfunctions of the Laplacian. One focus is on nodal and critical point sets of quantum ergodic eigenfunctions. C. Sogge and the principal investigator recently found an identity relating nodal set integrals to L1 norms of eigenfunctions, which was used to give a lower bound on volumes of nodal sets. The PI has proved an analogous identity for critical points and plan to use it to study the distribution of critical points. The PI also plans to continue a study of the analytic continuation of eigenfunctions to Grauert tubes as a tool to study complex zeros and critical points. Using a new result with J. Toth on quantum ergodic restriction to hypersurfaces, the PI plans to work out the distribution of intersections of geodesics and nodal sets. In Kaehler geometry, this project will develop the approximation theory of Kaehler metrics in a fixed class by Bergman metrics. In particular the PI is studying the initial value problem for geodesics in the space of Kaehler metrics in joint work with Y. Rubinstein. We have shown that there are obstructions to solving it for long times in most directions. Another project concerns "random metrics," i.e. probability measures on the space of Kaehler metrics which are defined by approximating it by the symmetric space of Bergman metrics. For almost 100 years now, quantum mechanics and quantum field theory are the basic theories in physics. Quantum physics is not an intuitive theory that can be related to the ordinary world in the way that mechanics or electricity and magnetism can be. So it is important to study simple models which exhibit the basic quantum features. Vibrating drums and surfaces have been studied since Chladni in 1800 and yet the basic questions asked by Chladni are completely open. If one pours sand on a vibrating drum, it will settle into "nodal patterns" which are the points where the vibrating drum is not moving up and down but staying still. These nodal patterns allow us to visualize the modes of vibration of the drum. As the frequency of vibration increases, the pattern becomes more and more complicated. There exist youtube videos of sand poured on vibrating drums where one can see the patterns, but it is very hard to predict what they will look like. Physicists conjecture that if the billiards played on the drum are "chaotic" (i.e unpredictable and essentially random), then the nodal lines will also be like random curves. The principal investigator has shown that if one extends the modes of vibration into the complex domain (i.e. the world of complex numbers) then the physics conjectures are true. The PI is now trying to prove the same along trajectories of the billiard balls. One also hopes to understand the "critical points," i.e., the points where the drum vibrates to its top and bottom positions. The same techniques apply to the much more abstruse physics of quantum gravity, and the principal investigator is collaborating with two physicists on that.