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 central conjecture of quantum scattering theory – the asymptotic completeness of quantum many-body systems. This property, previously verified only experimentally, says that left to its own devices a quantum system of many particles breaks up, after a period of time, into stable, independently moving fragments. ([129] [141])

2.    Quantum Theory of Radiation

Constructing, jointly with V. Bach and J. Fröhlich, the mathematical theory of the physical phenomenon which gave birth to quantum physics – emission and absorption of electro-magnetic radiation by systems of non-relativistic matter, such as atoms and molecules. ([55, 99, 101])

3.    Spectral Renormalization Group

Developing, jointly with V. Bach and J. Fröhlich, the novel spectral renormalization group (SRG) theory yielding among other things the perturbation theory of non-isolated eigenvalues. (The perturbation theory of isolated eigenvalues, an important area of mathematics pioneered in works of Schrödinger, Dirac and von Neumann in the 1920s, was developed over a period of about 50 years by F. Rellich, T. Kato, B. Simon and many others.) In application to the non-relativistic quantum electrodynamics (QED), the SRG gives the first rigorous and effective method of computation of radiative corrections (in particular, the Lamb shift) and life-times for atomic energies. ([55, 102])

   

Significant Recent Research Contributions

4.   Propagation of Quantum Information

Proof, jointly with J. Faupin and M. Lemm, of the long-standing conjecture on existence of an effective light cone in the paradigmatic Bose-Hubbard model of the condensed matter physics. Proof, jointly with Jingxuan Zhang, of general time constraints on various manifestations of flow of quantum information in general discrete quantum many-body systems. ([1, 5, 9])

5.   Dynamical Stability of Vortex Matter

Proof, jointly with T. Tzaneteas, of stability of Abrikosov vortex lattices (describing the ground states of superconductors at weak magnetic fields and at nanoscopic scales). ([24])

6.   Rayleigh Scattering of Electrons and Photons

Proof, jointly with J. Faupin, of asymptotic completeness of Rayleigh scattering (scattering of photons on atoms at low energies). ([31])

7.   Theory of Large Coulomb Systems

Proof, jointly with I. Anapolitanos, of the universal long distance behaviour of van der Waals force. ([25])

8.   Density Functional Theory

9.   Symmetry Breaking and Existence of Vortex Lattices

Proof, jointly with I. Chenn, A. Gardner and K. Rajaratnam, of the symmetry breaking and existence of vortex lattice solutions in:

10.  Neckpinching under Mean Curvature Flow

Proof, jointly with Z. Gang and D. Knopf, of stability of neckpinching of non-compact surfaces evolving under mean curvature flow. ([22])

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Page last updated: December 2022