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I. M. Sigal

Publications and source records attributed to I. M. Sigal.

At least 19 recordsLinked to original sources
arXiv2019

Ginzburg-Landau equations on Riemann surfaces of higher genus

We study the Ginzburg-Landau equations on Riemann surfaces of arbitrary genus. In particular: - we construct explicitly the (local moduli space of gauge-equivalent) solutions in a neighbourhood of the constant curvature ones; - classify holomorphic structures on line bundles arising as solutions to the equations in terms of the degree, the Abel-Jacobi map, and symmetric products of the surface; - determine the form of the energy and identify when it is below the energy of the constant curvature (normal) solutions.

math.AP↗
arXiv2013

Renormalization Group and Problem of Radiation

The standard model of non-relativistic quantum electrodynamics describes non-relativistic quantum matter, such as atoms and molecules, coupled to the quantized electromagnetic field. Within this model, we review basic notions, results and techniques in the theory of radiation. We describe the key technique in this area - the spectral renormalization group. Our review is based on joint works with Volker Bach and Juerg Froehlich and with Walid Abou Salem, Thomas Chen, Jeremy Faupin and Marcel Griesemer. Brief discussion of related contributions is given at the end of these lectures. This review will appear in "Quantum Theory from Small to Large Scales", Lecture Notes of the Les Houches Summer Schools, volume 95, Oxford University Press, 2011. Key words: quantum electrodynamics, photons and electrons, renormalization group, quantum resonances, spectral theory, Schroedinger operators, ground state, quantum dynamics, non-relativistic theory.

math-ph↗
arXiv2013

On blowup dynamics in the Keller-Segel model of chemotaxis

We investigate the (reduced) Keller-Segel equations modeling chemotaxis of bio-organisms. We present a formal derivation and partial rigorous results of the blowup dynamics of solution of these equations describing the chemotactic aggregation of the organisms. Our results are confirmed by numerical simulations and the formula we derive coincides with the formula of Herrero and Velázquez for specially constructed solutions.

nlin.PS↗
arXiv2011

On Abrikosov Lattice Solutions of the Ginzburg-Landau Equations

Building on earlier work, we have given in our paper in Contemporary Mathematics 535, 195-213, 2011 (referred here as [TS]) a proof of existence of Abrikosov vortex lattices in the Ginzburg-Landau model of superconductivity and have shown that the triangular lattice gives the lowest energy per lattice cell. After [TS] was published, we realized that it proves a stronger result than was stated there. This result is recorded in the present paper. The proofs remain the same as in [TS], apart from some streamlining.

math-ph↗
arXiv2011

On Blowup in Nonlinear Heat Equations

We establish the asymptotics of blowup for nonlinear heat equations with superlinear power nonlinearities in arbitrary dimensions and we estimate the remainders.

math.AP↗
arXiv2011

On Spectra of Linearized Operators for Keller-Segel Models of Chemotaxis

We consider the phenomenon of collapse in the critical Keller-Segel equation (KS) which models chemotactic aggregation of micro-organisms underlying many social activities, e.g. fruiting body development and biofilm formation. Also KS describes the collapse of a gas of self-gravitating Brownian particles. We find the fluctuation spectrum around the collapsing family of steady states for these equations, which is instrumental in derivation of the critical collapse law. To this end we develop a rigorous version of the method of matched asymptotics for the spectral analysis of a class of second order differential operators containing the linearized Keller-Segel operators (and as we argue linearized operators appearing in nonlinear evolution problems). We explain how the results we obtain are used to derive the critical collapse law, as well as for proving its stability.

nlin.PS↗
arXiv2011

Stability of Abrikosov lattices under gauge-periodic perturbations

We consider Abrikosov-type vortex lattice solutions of the Ginzburg-Landau equations of superconductivity, consisting of single vortices, for magnetic fields below but close to the second critical magnetic field H_{c2} = kappa^2 and for superconductors filling the entire R^2. Here kappa is the Ginzburg-Landau parameter. The lattice shape, parameterized by tau, is allowed to be arbitrary (not just triangular or rectangular). Within the context of the time-dependent Ginzburg-Landau equations, called the Gorkov-Eliashberg-Schmidt equations, we prove that such lattices are asymptotically stable under gauge periodic perturbations for kappa^2 > (1/2)(1 - (1/beta(tau)) and unstable for kappa^2 < (1/2)(1 - (1/beta(tau)), where beta(tau) is the Abrikosov constant depending on the lattice shape tau. This result goes against the common belief among physicists and mathematicians that Abrikosov-type vortex lattice solutions are stable only for triangular lattices and kappa^2 > 1/2. (There is no real contradiction though as we consider very special perturbations.)

math-ph↗
arXiv2009

Resonant Perturbation Theory of Decoherence and Relaxation of Quantum Bits

We describe our recent results on the resonant perturbation theory of decoherence and relaxation for quantum system with many qubits. The approach represents a rigorous analysis of the phenomenon of decoherence and relaxation for general $N$-level systems coupled to reservoirs of the bosonic fields. We derive a representation of the reduced dynamics valid for all times $t\geq 0$ and for small but fixed interaction strength. Our approach does not involve master equation approximations and applies to a wide variety of systems which are not explicitly solvable.

quant-ph↗
arXiv2009

Local Decay in Non-relativistic QED

We prove the limiting absorption principle for a dressed electron at a fixed total momentum in the standard model of non-relativistic quantum electrodynamics. Our proof is based on an application of the smooth Feshbach-Schur map in conjunction with Mourre's theory.

math-ph↗
arXiv2009

On collapse of wave maps

We derive the universal collapse law of degree 1 equivariant wave maps (solutions of the sigma-model) from the 2+1 Minkowski space-time,to the 2-sphere. To this end we introduce a nonlinear transformation from original variables to blowup ones. Our formal derivations are confirmed by numerical simulations.

math.AP↗
arXiv2009

Ground State and Resonances in the Standard Model of Non-relativistic QED

We prove existence of a ground state and resonances in the standard model of the non-relativistic quantum electro-dynamics (QED). To this end we introduce a new canonical transformation of QED Hamiltonians and use the spectral renormalization group technique with a new choice of Banach spaces.

math-ph↗
arXiv2008

Colliding solitons for the nonlinear Schrodinger equation

We study the collision of two fast solitons for the nonlinear Schrödinger equation in the presence of a spatially adiabatic external potential. For a high initial relative speed $\|v\|$ of the solitons, we show that, up to times of order $\log\|v\|$ after the collision, the solitons preserve their shape (in $L^2$-norm), and the dynamics of the centers of mass of the solitons is approximately determined by the external potential, plus error terms due to radiation damping and the extended nature of the solitons. We remark on how to obtain longer time scales under stronger assumptions on the initial condition and the external potential.

math-ph↗
arXiv2008

Dynamics of Collective Decoherence and Thermalization

We analyze the dynamics of N interacting spins (quantum register) collectively coupled to a thermal environment. Each spin experiences the same environment interaction, consisting of an energy conserving and an energy exchange part. We find the decay rates of the reduced density matrix elements in the energy basis. We show that if the spins do not interact among each other, then the fastest decay rates of off-diagonal matrix elements induced by the energy conserving interaction is of order N^2, while that one induced by the energy exchange interaction is of the order N only. Moreover, the diagonal matrix elements approach their limiting values at a rate independent of N. For a general spin system the decay rates depend in a rather complicated (but explicit) way on the size N and the interaction between the spins. Our method is based on a dynamical quantum resonance theory valid for small, fixed values of the couplings. We do not make Markov-, Born- or weak coupling (van Hove) approximations.

math-ph↗
arXiv2008

On the theory of resonances in non-relativistic QED and related models

We study the mathematical theory of quantum resonances in the standard model of non-relativistic QED and in Nelson's model. In particular, we estimate the survival probability of metastable states corresponding to quantum resonances and relate the resonances to poles of an analytic continuation of matrix elements of the resolvent of the quantum Hamiltonian.

math-ph↗