20 apr 2004

Abstract.

Seminario di Analisi

when: Martedi' 20 aprile, ore 18.00 Tuesday, April 20, at 6 pm:

title: Sulla stabilita asintotica degli stati fondamentali dell'equazione di Schrodinger nonlineare On the asymptotic stability of fundamental; states of nonlinear Schrodinger equation

abstract: During the early 80's, M.Weinstein and J.Shatah proved independently that, roughly, linear stability (that is, position of eigenvalues of a specific linearized operator) implies nonlinear orbital stability. Weinstein and Shatah used essentilly ODE methods, leaving open the question of asymptotic stability.

Various authors have studied recently the dispersive behaviour of the error terms by focusing on the linearization of the problem. There is a fairly good understanding about the behaviour of the continuous modes, thanks to technology from the theory of Schrodinger operators, that we have adapted to the problem at hand. It is very hard to locate discrete modes and study their dynamics and the interaction between discrete modes and continuous ones. If however discrete modes correspond to eigenvalues close to the continuous spectrum, the nonlinear coupling between continuous and discrete modes dampens the latter ones. This mechanism is called Nonlinear Fermi Golden Rule and was first introduced  by I.M.Sigal.

In the present talk we will present the problem, sketch the theory of the continuous spectrum and, time permitting, give a sketch of the Fermi Golden Rule.

posted by G. Alberti