Semiclassical Spectra and Pseudospectra for non-self-adjoint operators.
Abstract of Proposed Research Michael Hitrik
This research project is concerned with the spectral analysis of classes of non-self-adjoint linear differential operators in the semiclassical limit, both in the Euclidean and compact settings. Specifically, the proposer plans to undertake a detailed study of spectra for non-self-adjoint perturbations of self-adjoint semiclassical operators, in the case when the classical Hamilton flow of the unperturbed part is completely integrable or is a small perturbation thereof. The main tool for understanding spectral contributions coming from flow invariant Lagrangian tori is the construction of quantum analogues of the classical Birkhoff normal form near the tori, analyzed using the techniques of FBI--Bargmann transformations. In a related project, the proposer seeks to understand the precise location of scattering poles for strictly convex analytic obstacles in dimensions 2 and 3, with the aim of describing asymptotics of the individual poles by means of Bohr-Sommerfeld type quantization conditions. Another direction of research is concerned with the tunnel effect and return to thermodynamical equilibrium of Kramers-Fokker-Planck operators. Random perturbations of non-self-adjoint operators as well as the distribution of resonances for semiclassical Schrodinger operators will also be studied in this project.
For a broad range of physical phenomena, where dissipation or propagation to infinity are possible, a state is described by its rate of oscillation as well as its rate of decay. This information is encoded in sequences of complex numbers identified as eigenvalues of a non-self-adjoint operator corresponding to the physical problem in question. For instance, resonances associated to quantum mechanical systems, such as unstable molecules, and appearing naturally also in electromagnetic and acoustic scattering, may be viewed this way. Asymptotic analysis of these quantities in the high energy or semiclassical regime is very rich and challenging mathematically, revealing fundamental and subtle links between quantum objects (complex eigenvalues or resonances) and corresponding classical objects, such as periodic/trapped trajectories or invariant tori. The proposed research thus seeks to shed some new light on universal relations between the classical and quantum views of the world. Closer to applications, a better understanding of the distribution of resonances for convex bodies may help advance the inverse problem of determining characteristics of a scatterer from its resonant frequencies in nondestructive testing technologies.
