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В четверг, 2 ноября 2006 года, в 15:40 в конференц-зале НМУ, Б. Власьевский 11, состоится доклад: Nonlinear hamiltonian pde and quantum mechanics. Лектор – Александр Комеч (МГУ).

All basic equations of Quantum Mechanics are linear Hamiltonian PDEs: the Schrodinger, Dirac, Yang-Mills eqn etc. On the other hand, the autonomous linear equations (with static potentials) are not compatible with basic quantum phenomena:

A: Transitions between quantum stationary states, and

B: Wave-particle duality.

A selfconsistent description of the phenomena needs the coupling to the Maxwell field since the transitions are followed by the electromagnetic radiation which is a solution to the Maxwell eqns. The coupling is nonlinear (it is the product of the wave function by the Maxwell potentials), that is known since the first Schroedinger papers (1926).

Hence, the questions arise

I. On a mathematical treatment of the phenomena as inherent properties of the coupled Maxwell-Schroedinger and Maxwell-Dirac eqns. We suggest the following treatment:

A: Global attraction to the set of «stationary states»;

B: Soliton type asymptotics.

II. On a universal character of the phenomena in the context of general nonlinear Hamiltonian PDEs.

We will present the results for nonlinear Hamiltonian PDEs on the global attractors and soliton type asymptotics obtained in last decade.

In particular, the global attractors and definition of «stationary states» depend on the symmetry Lie group of the eqn.