Semi-Classical Analysis

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

PI: Maciej R. Zworski, UC-Berkeley. DMS-0200732 Abstract: The main interest of the PI is the study of quantum mechanics from the mathematical point of view, and of its many manifestations in the theory of partial differential equations and geometry. Specific current interests are the classical/quantum correspondence, resonances, geometric scattering, and non-hermitian quantum mechanics. More precisely, the PI is interested in resonances, which are mathematical objects modeling states which have certain frequencies of oscillations (or rest energies) and rates of decay, such as unstable molecules or classical system responding to resonant forcing terms. Despite a long tradition and a lot of recent progress our understanding is still very limited. Current experimental and numerical advances provide new stimuli for our studies. Another interest of the PI is non-hermitian quantum mechanics, which deals with systems in which energy is not conserved. That is almost always present when we localize a part of a system and the global conservation of energy disapears. In a subtle way resonances already fall into this category of phenomena. The mathematical problems present here are the stability of eigenvalues and measuring the size of the resolvent of non-self-adjoint operators. That leads to the study of ``pseudospectra'' which is then related to many interesting phenomena in PDEs. Finally, the PI's interest involves mathematical scattering. It replaces spectral theory for problems on non-compact domains, and in physics almost all the data comes from scattering experiments. Many new things are constantly discovered now, ranging from scattering on locally symmetric spaces to problems related to conformal field theory. Many of the phenomena discussed in this proposal are in fact more general: for instance, electromagnetic scattering can be used to model quantum scattering, and its understanding can benefit from the development of the classical/quantum correspondence. The PI's work focuses on the search for general mathematical principles, and the detailed study of specific examples is motivated by that.