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(references are given in the List of Publications)
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])
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])
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])
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])
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])
Proof, jointly with J. Faupin, of asymptotic completeness of Rayleigh scattering (scattering of photons on atoms at low energies). ([31])
Proof, jointly with I. Anapolitanos, of the universal long distance behaviour of van der Waals force. ([25])
Proof, jointly with I. Chenn, A. Gardner and K. Rajaratnam, of the symmetry breaking and existence of vortex lattice solutions in:
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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