Institute of Mathematics

Talk

Modul:   MAT673  Seminar PDE and Mathematical Physics

Asymptotic quantization with an exponentially small error

Prof. Dr. Gueorgui Popov talk

Speaker invited by: Prof. Dr. Thomas Kappeler

Date: 12.03.15  Time: 18.15 - 19.15  Room: Y27H35/36

The aim of this talk is to provide a construction of different types of quasi-modes with an exponentially small discrepancy for semi-classical formally selfadjoint differential operators with (analytic) Gevrey coefficients in an (analytic) Gevrey smooth manifold of dimension greater or equal to two. There are three types of quasi-modes we are interested in which are distinguished by their micro-supports - quasi-modes associated with a given Kronecker invariant torus of the classical Hamiltonian with a Diophantine vector of rotation, quasi-modes associated with a large family of such tori in any dimension and the so called Shnirelman quasi-modes associated with gaps between invariant tori when the dimension of the manifold is two. The construction is based on Birkhoff and Quantum Birkhoff normal forms in Gevrey classes and on the calculus of Fourier Integral Operators in Gevrey classes. An analogue of the effective stability in the semi-classical limit is obtained as well.