Israel Michael Sigal

Most Significant Research Contributions

(references are given in the List of Publications)

1.    Scattering Theory

Proof (jointly with A. Soffer) of the conjecture of the asymptotic completeness of many-body systems. The asymptotic completeness - the central mathematical statement of quantum scattering theory - asserts that left to its own devices a system of many particles after a period of time breaks up into stable, independently moving fragments.

2.    Quantum Theory of Radiation

Constructing, jointly with V. Bach and J. Fröhlich, the mathematical theory of quantum radiation processes. The latter addresses the physical phenomenon standing at the origin of quantum theory - emission and absorption of radiation by systems of non-relativistic matter, such as atoms and molecules. The mathematical theory mentioned above gives the first consistent and effective method of computation of radiative corrections (in particular, the Lamb shift) and life-times.

3.    Theory of Vortex Motion

(a) Proof, jointly with S. Gustafson, of the long-standing conjecture of Jaffe and Taubes that in type I superconductors the magnetic forties are stable for any vorticity n, while in type II superconductors they are stable for |n| = 1 and unstable |n| > 1.
(b) Derivation, jointly with S. Gustafson, of magnetic vortex dynamics for the abelean Higgs model in particle physics and for the Gorkov-Eliashberg model for superconductors. These results show that initially separated vortices evolve for long time intervals as rigid objects parameterized by their centers and phases.
(c) Proof, jointly with T. Tzaneteas, of existence and stability of Abrikosov magnetic vortex lattices.

4.    Rayleigh Scattering of Electrons and Photons

Proof, jointly with J. Faupin, of asymptotic completeness of Rayleigh scattering, i.e. of scattering of photons on atoms at low energies.

5.    Theory of Large Coulomb Systems

(a) Proof of instability of large negative ions (saturation of binding - a given nucleus can bind only finite number of electrons);
(b) Proof, jointly with V. Ivrii, of the Scott Conjecture regarding the behaviour of ground states of large molecules.

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