The transmission boundary condition which is considered appears in various exchange problems such as molecular diffusion across semi-permeable membranes (Tanner 1978, Novikov 1998), heat transfer between two materials (Carslaw, Gilkey, Bardos), or transverse magnetization evolution in nuclear magnetic resonance (NMR) experiments (Grebenkov 2010). In the simplest setting of the latter case, one considers the local transverse magnetization G(x,y;t)produced by the nuclei that started from a fixed initial point y and diffused in a constant magnetic field gradient g up to time t. This magnetization is also called the propagator or the Green function of the Bloch-Torrey equation (Torrey 1956). In this talk, we focus on the one-dimensional situation, in which the operator -d^2/dx^2+ix is called the complex Airy operator and appears in many contexts: mathematical physics, fluid dynamics, time dependent Ginzburg-Landau problems and also as an interesting toy model in spectral theory (see Almog).
Biography
Professor Bernard Helffer is now Professor Emeritus at the University of Nantes and associate researcher at the University of Paris-Sud. From June 2010 to June 2012, he had been President of the Société Mathématique de France. His research interests are: partial differential equations, analysis on nilpotent Lie groups, spectral theory, semi-classical analysis, mathematical Physics, statistical mechanics, superconductivity and liquid crystals, nodal sets and minimal partitions.