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(references are given in the List of Publications)
Proof (jointly with A. Soffer) of the N-body asymptotic completeness. The asymptotic completeness states that left to its own devices a system of many particles after a period of time breaks up into stable, independently moving fragments. Main work: N-particle scattering problem: asymptotic completenesss for short-range systems, Annal. Math. 125 (1987) 35-108 (with A. Soffer).
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. Main work: Quantum electrodynamics of confined non-relativistic particles, Adv. Math., 137 (1998), 205-298 (with V. Bach and J. Fröhlich).
Proof of instability of time-periodic, spacially localized solutions of wave equations and related equations under arbitrary weak nonlinear perturbations; establishing jointly with R. Pyke general limitations on periods of time-periodic solutions on nonlinear wave equations and related equations Main work: Non-Linear wave and Schrodinger equations I, instability of periodic and quasiperiodic solutions, Comm. Math. Phys. 153 (1993) 297-320; Nonlinear wave equations: Constraints on periods and exponential bounds for periodic solutions, Duke Math. J., 88 (1997), 133-180 (with R. Pyke).
Developing, jointly with V. Bach and J. Fröhlich, the Renormalization-group approach to spectral and dynamical problems including introducing the new concept of spectral renormalization-group flow (acting directly on space of equations).
Development, jointly with P. Hislop and simultaneously with B. Helffer & J. Sjoestrand and J.-M. Combes, P. Duclos, M. Klein & R. Seiler, of the mathematical theory of tunneling resonances, one of the major constructs of quantum physics, underpinning such phenomena as nuclear instability. Main work: Shape resonances, A Memoir AMS, 78 N399, 1989 (with P. Hislop).
Proof os instability of large negative ions (saturation of binding - a given nucleus can bind only finite number of electrons); proof jointly with V. Ivrii of the Scott Conjecture regarding the behavior of ground states of large molecules. Main work: Asymptotics of the ground state energies of Large Coulomb systems, Annals of Mathematics 138 (1993), 243-335 (with V. Ivrii).
Proof, jointly with S. Gustafson, of the long-standing conjecture of Jaffe and Taubes that in type I superconductors the magnetic vortices are stable for any vorticity n, while in type II superconductors they are stable for |n| = 1 and unstable for |n| > 1. Derivation, jointly with S. Gustafson, effective dynamical law for multi-vortex systems. (The 2003 Nobel prize in Physics was shared by A. A. Brikosov who found vortetx solutions of the Ginzburg-Landau equation). Descriptionjointly with J. Frölich, S. Gustafson and L. Jonsson dynamics of solitons in nonlinear Schrödinger equations with external potentials. Proof, jointly with Gang Zhou, of relaxation of solitons in an external potential : On soliton dynamics in nonlinear Schrödinger equations, CAFA (2006), together (with Zhou Gang). Main work: The stability of magnetic vortices, Comm. Math. Phys. 210 (2000), 257-276 (with S. Gustafson) Dynamics of magnetic vortices, Advances in Mathematics 199 (2006) (448-498) (with S. Gustafson). Dynamics of solitary waves in external potentials, Commun Math Phys 250 (2004) (with J. Fröhlich, S. Gustafson and L. Jonsson).
Proof, jointly with V. Bach and J. Fröhlich, of the property return to equilibrium for quantum systems coupled to thermal reservoirs at all temperatures (this property states that a quantum system disturbed from its state of thermal equilibrium returns tot that state with the progress of time; proof, jointly with M. Merkli and M. Mück existence and dynamical stability of non-equilibrium stationary states carrying constant fluxes and having positive entropy production. Main work: Return to equilibrium, J. Math. Phys 41 (2000) 3985-4060 (with V. Bach and J. Fröhlich); Theory of non-equilibrium stationary states a theory of resonances (preprint, 2004) (with M. Merkli and M. Mück).