**Organizers**

- Vladimir Buslaev, St. Petersburg University
- Andrew Comech, IITP RAN and Texas A&M University
- Dmitrii Kazakov, Joint Institute of Nuclear Research (Dubna) and ITEP
- Alexander Komech, IITP RAN and University of Vienna
- Alexander Kuleshov, Institute for Information Transmission Problems, RUSSIA
- Robert Minlos, Institute for Information Transmission Problems, RUSSIA
- Sergey Vergeles, L.D.Landau Institute for Theoretical Physics RAN and MFTI

**Contact information**

Andrew Comech (comech@gmail.com), Alexander Komech (akomech@iitp.ru)

**Description**

The aims of the research semester are the long-time asymptotics and the scattering problems of Quantum Mechanics and Quantum Electrodynamics in a general context of nonlinear Hamiltonian partial differential equations. The core of the research semester consists of several lecture courses which will cover the basics of attractors and quantum scattering theory from the point of view of Partial Differential Equations and Quantum Physics.

We are going to discuss recent progress and unsolved problems. We also plan a wide discussion of related mathematical problems of Classical and Quantum Physics, such as scattering cross section, S-matrix, Feynman diagrams and renormalization, convergence to stationary states and solitary waves, solitary asymptotics, and the correspondence between the structure of the global attractor and the symmetry group of the dynamical equation.

The mathematical tools of the research area are rooted in Complex, Harmonic, and Functional Analysis, Spectral Theory, and Scattering theory for Partial Differential Equations. The physical background is provided by the methods of Quantum Mechanics and Quantum Electrodynamics: Feynman diagrams, functional integral, and renormalization.

This interconnection of several disciplines allows to engage top specialists from adjacent fields, who represent the main body of the group. Their collaboration started in the focused research group of A.I.Komech at the Faculty of Mathematics of the University of Vienna in 2002-2005 and continued in IITP RAN in 2005-2008, BIRS focused research group in May 2007, and Oberwolfach miniworkshop in February 2008.

**Program**

- Moscow, 15.12.2008 - 15.01.2009
Workshop "Mathematical Problems of Quantum Mechanics"

Lecturers: A.I. Komech and D. Kazakov

- Moscow, 1.02.2009 - 28.02.2009
Workshop "Mathematical Problems of Quantum Electrodynamics"

Lecturers: A.I. Komech and S. Vergeles

- Vienna, 1.03.2009 - 30.04.2009
School "Nonrelativistic Quantum Scattering Theory"

Lecturers: A.I. Komech and E. Kopylova

- Moscow, 1.05.2009 - 30.05.2009
Workshop "Relativistic Quantum Scattering Theory"

Lecturers: A.I. Komech and E. Kopylova

- Vilnius, 1.07.2009 - 31.07.2009
School/Workshop "Attractors of Nonlinear Hamiltonian PDEs"

Lecturers: A.I. Komech, E. Kopylova and V. Imaykin

**Lecture Courses:**

**I. Lecturer:** A.I. Komech

**Title: Nonrelativistic and Relativistic Quantum Mechanics**

**Programme:**

1. Origine of the quantum mechanics: spectroscopy, black body emission, photoeffect, the Bohr-Sommerfeld ``Old Quantum Mechanics'', L. de Broglie' wave-particle duality, quasiclassical asymptotics for oscillatory initial data, the Schrödinger equation.

2. Hamiltonian and Lagrangian formalism for the Schrödinger equation, conservation laws (charge, current, momentum, anglular momentum, the Noether theorem on currents).

3. The Heisenberg matrix mechanics.

4. Scattering of light and electrons.

5. Normal Zeemann effect.

6. Spin: Stern-Gerlach double splitting and anomalous Zeemann effect, Pauli matrices and Pauli equation.

7. Special relativity and covariant electrodynamics.

8. The Dirac equation: the Dirac matrices, the Pauli theorem,

9. Hamiltonian and Lagrangian formalism for the Dirac equation, conservation laws (charge, current, momentum, anglular momentum).

**II. Lecturer:** D.I. Kazakov

**Title: Nonrelativistic and Relativistic Quantum Electrodynamics**

**Programme:**

1. The Fock space and second quantization for the free Schrödinger, Klein-Gordon and Dirac equations.

2. S-matrix: interaction representation and the Born series.

3. The Feynman diagramms.

4. Scattering cross section, Compton scattering and the Klein-Nishina-Tamm formula.

5. Anomalous magnetic moment and charge renormalization.

6. Lamb shift and the mass renormalization.

**A list of potential participants
and their affiliation
**

- Vladimir Buslaev, St. Petersburg University, RUSSIA
- Andrew Comech, Institute for Information Transmission Problems, RUSSIA
- Valery Imaikin, Town of Korolev, Moscow Region, 141080 RUSSIA
- Alexander Komech, Institute for Information Transmission Problems, RUSSIA
- Dmitrii Kazakov, Joint Institute of Nuclear Research in Dubna, RUSSIA
- Elena Kopylova, Institute for Information Transmission Problems, RUSSIA
- Boris Vainberg, UNC - Charlotte, USA
- Sergey Vergeles, L.D.Landau Institute of Theoretical Physics, RUSSIA