Scattering Theory

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DMS-9970614The main interest of the PI is the study of resonances and of the wave equation. Resonances which are described by complex numbers constitute a replacement of eigenvalues for problems on noncompact domains and they appear naturally in many branches of mathematics and physics. The real part of a resonance describes the energy (or frequency) of a state and the imaginary part its rate of decay. This constitutes a more realistic model than an eigenvalue which provides energy only and assumes eternal existence of a state. A new wealth of phenomena appears and our understanding is still rather fragmentary. Linear and non-linear wave equations describe many physical phenomena often closely related to resonances. In the contexts of propagation, spectral and scattering theories the PI will investigate them further. The specific directions include: understanding of the dynamical definition of resonances and their appearance in the long time behaviour of solutions of the wave and Schroedinger equations, obtaining lower bounds on the number of resonances in terms of dynamical quantities, developing the theory of the FBI transformation on non-compact manifolds with applications to resonances and to the study of propagation for Schroedinger equation in mind, estimates of resonances at low energies, understanding of "quantum chaos" in scattering theory.For a broad range of phenomena, a physical state can be described by twoparameters: its rest energy and its rate of decay. This information is elegantly encoded in a complex number, whose real part is the energy and the imaginary part, the rate of decay. This description of a state, called a resonance, appears naturally in mathematics, physics and chemistry: from the Riemann zeta function to experimental scattering data.The PI studies general principles in the distribution of resonances,their relation to wave propagation, and their behaviour in various specificsituations. He is also interested in "quantum chaos", the study of which raises questions in many areas: from number theory to mezoscopic systems of physics. The main issues are universal relations between the classical and quantum views of the world -- one of the yet unresolved central themes of the 20th century.