Research Highlights

Most Significant Research Contributions

(references are given in the List of Publications)

1. Scattering Theory

Proof, jointly with A. Soffer, of the asymptotic completeness of quantum many-body systems. This central conjecture of quantum scattering theory, 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. (139, 151)

2. Quantum Theory of Radiation

Constructing, jointly with V. Bach and J. Fröhlich, the mathematical theory of electro-magnetic radiation by systems of non-relativistic matter, the physical phenomenon which gave birth to quantum physics. (111, 109, 122)

3. Operator 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. (111, 112)

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 quantum lattice many-body lattice systems. Establishing the existence of the effective light cone in propagation of entanglement. (16, 7, 19)

5. Stability of Quantum Vortex Matter

Proof, jointly with T. Tzaneteas, of stability of Abrikosov vortex lattices (describing the ground states of superconductors at strong magnetic fields and at nanoscopic scales) and, jointly with S. Gustafson, stability of quantum vortices. (34)

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). (41)

7. Theory of Large Coulomb Systems

Proof, jointly with I. Anapolitanos, of the universal long-distance behaviour of the van der Waals force. (35)

8. Density Functional Theory

• Construction, jointly with F. Pusateri, of scattering theory for the short-range (scattering subcritical) correlation (nonlinear) terms for the Kohn-Sham equation (DFT). (21)
• Derivation, jointly with I. Chenn, of the classical Poisson equation in a dielectric medium at positive temperatures from the microscopic reduced Hartree-Fock equation. (25)

9. 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. (32)

10. Dynamics of Solitons

• Proof, jointly with Zhou Gang, of relaxation of solitons in generalized Schrödinger equations with potentials. It is shown that the solitons relax to stationary solitons trapped near minima of the potentials, with the centers obeying Newton's law with a friction term due to the outgoing radiation (the Langevin equation).
• Generalization, jointly with Fröhlich, Gustafson and Jonsson, of the Bronski-Jerrard and Fröhlich-Tsai-Yau results on the Newtonian dynamics for solitons in the nonlinear Schrödinger equations with potentials and introduction of a new geometrical technique to this area.
• Derivation, jointly with Bizon and Ovchinnikov and with Ovchinnikov, of scaling dynamical law of collapse in the Yang-Mills and the wave map equations.
• Derivation, jointly with S. Dejak, of (the first in the literature) effective dynamics of solitons in KdV and generalized KdV equations for channels with variable bottoms.

11. 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:

• the Bardeen-Cooper-Schrieffer theory of superconductivity (the Bogolubov-de Gennes equations); (27)
• the Weinberg-Salam theory of electroweak interactions (the vacuum Weinberg-Salam, or Yang-Mills-Higgs, equations); (11)
• the fractional quantum Hall effect (the Jackiw-Pi and Zhang-Hansen-Kivelson equations). (26)

Page last updated: September 2026