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I. M. Sigal

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

34 records · Page 2Linked to original sources
arXiv2007

Spectral Theory for the Standard Model of Non-Relativistic QED

For a model of atoms and molecules made from static nuclei and non-relativistic electrons coupled to the quantized radiation field (the standard model of non-relativistic QED), we prove a Mourre estimate and a limiting absorption principle in a neighborhood of the ground state energy. As corollaries we derive local decay estimates for the photon dynamics, and we prove absence of (excited) eigenvalues and absolute continuity of the energy spectrum near the ground state energy, a region of the spectrum not understood in previous investigations. The conjugate operator in our Mourre estimate is the second quantized generator of dilatations on Fock space.

math-ph↗
arXiv2007

Theory of Non-Equilibrium Sationary States as a Theory of Resonances

We study a small quantum system (e.g. a simplified model for an atom or molecule) interacting with two bosonic or fermionic reservoirs (say, photon or phonon fields). We show that the combined system has a family of stationary states, parametrized by two numbers $T_1$, $T_2$ (``reservoir temperatures''). If $T_1\neq T_2$, then these states are non-equilibrium, stationary states (NESS). In the latter case we show that they have nonvanishing heat fluxes and positive entropy production. Furthermore, we show that these states are dynamically asymptotically stable. The latter means that the evolution with an initial condition, normal with respect to any state where the reservoirs are in equilibria at temperatures $T_1$ and $T_2$, converges to the corresponding NESS. Our results are valid for the temperatures satisfying the bound $\min(T_1, T_2) > g^{2+α}$, where $g$ is the coupling constant and $0< α<1$ is a power related to the infra-red behaviour of the coupling functions.

math-ph↗
arXiv2007

Resonance Theory of Decoherence and Thermalization

We present a rigorous analysis of the phenomenon of decoherence for general $N-$level systems coupled to reservoirs. The latter are described by free massless bosonic fields. We apply our general results to the specific cases of the qubit and the quantum register. We compare our results with the explicitly solvable case of systems whose interaction with the environment does not allow for energy exchange (non-demolition, or energy conserving interactions). We suggest a new approach which applies to a wide variety of systems which are not explicitly solvable.

quant-ph↗
arXiv2006

Blow-up in Nonlinear Heat Equations

We study the blow-up problem of one-dimensional nonlinear heat equations. Our result shows that for a certain class of initial conditions, the solutions blow up in finite time and we characterize the asymptotic dynamics of these solutions. The approach is analogous to one used in bifurcation theory and our techniques can be regarded as a time-dependent version of the Lyapunov-Schmidt decomposition.

math.AP↗
arXiv2006

Decoherence and Thermalization

We present a rigorous analysis of the phenomenon of decoherence for general N-level systems coupled to reservoirs of free massless bosonic fields. We apply our general results to the specific case of the qubit. Our approach does not involve master equation approximations and applies to a wide variety of systems which are not explicitly solvable.

quant-ph↗
arXiv2006

Relaxation of Solitons in Nonlinear Schrodinger Equations with Potential

In this paper we study dynamics of solitons in the generalized nonlinear Schrödinger equation (NLS) with an external potential in all dimensions except for 2. For a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space, centered near different critical points of the potential. We call those solutions which are centered near the minima of the potential and which minimize energy restricted to $\mathcal{L}^2-$unit sphere, trapped solitons or just solitons. In this paper we prove, under certain conditions on the potentials and initial conditions, that trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton's center and momentum) is close to Newton's equation but with a dissipative term due to radiation of the energy to infinity.

math-ph↗
arXiv2006

On Soliton Dynamics in Nonlinear Schrodinger Equations

In this paper we announce the result of asymptotic dynamics of solitons of nonlinear Schrodinger equations with external potentials. To each local minima of the potential there is a soliton centered around it. Under some conditions on the nonlinearity, the potential and the datum, we prove that the solution can be decomposed into two parts: the soliton and the term dissipating to infinity.

math-ph↗
arXiv2005

Instability of Equilibrium States for Coupled Heat Reservoirs at Different Temperatures

We consider quantum systems consisting of a ``small'' system coupled to two reservoirs (called open systems). We show that such a system has no equilibrium states normal with respect to any state of the decoupled system in which the reservoirs are at different temperatures, provided that either the temperatures or the temperature difference divided by the product of the temperatures are not too small. Our proof involves an elaborate spectral analysis of a general class of generators of the dynamics of open quantum systems, including quantum Liouville operators (``positive temperature Hamiltonians'') which generate the dynamics of the systems under consideration.

math-ph↗
arXiv2005

Long time motion of NLS solitary waves in a confining potential

We study the motion of solitary-wave solutions of a family of focusing generalized nonlinear Schroedinger equations with a confining, slowly varying external potential, $V(x)$. A Lyapunov-Schmidt decomposition of the solution combined with energy estimates allows us to control the motion of the solitary wave over a long, but finite, time interval. We show that the center of mass of the solitary wave follows a trajectory close to that of a Newtonian point particle in the external potential $V(x)$ over a long time interval.

math-ph↗
arXiv2004

Long-Time Dynamics of KdV Solitary Waves over a Variable Bottom

We study the variable bottom generalized Korteweg-de Vries (bKdV) equation dt u=-dx(dx^2 u+f(u)-b(t,x)u), where f is a nonlinearity and b is a small, bounded and slowly varying function related to the varying depth of a channel of water. Many variable coefficient KdV-type equations, including the variable coefficient, variable bottom KdV equation, can be rescaled into the bKdV. We study the long time behaviour of solutions with initial conditions close to a stable, b=0 solitary wave. We prove that for long time intervals, such solutions have the form of the solitary wave, whose centre and scale evolve according to a certain dynamical law involving the function b(t,x), plus an H^1-small fluctuation.

math-ph↗
arXiv2003

Effective Dynamics of Magnetic Vortices

We study solutions of Ginzburg-Landau-type evolution equations (both dissipative and Hamiltonian) with initial data representing collections of widely-spaced vortices. We show that for long times, the solutions continue to describe collections of vortices, and we identify (to leading order in the vortex separation) the dynamical system describing the motion of the vortex centres (effective dynamics).

math.AP↗
arXiv2003

Solitary Wave Dynamics in an External Potential

We study the behavior of solitary-wave solutions of some generalized nonlinear Schrödinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves. We construct solutions of the equations with a non-vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that, over a large interval of time, they describe a solitary wave whose center of mass motion is a solution of Newton's equations of motion for a point particle in the given external potential, up to small corrections corresponding to radiation damping.

math-ph↗
arXiv2003

Collapse of an Instanton

We construct a two parameter family of collapsing solutions to the 4+1 Yang-Mills equations and derive the dynamical law of the collapse. Our arguments indicate that this family of solutions is stable. The latter fact is also supported by numerical simulations.

math-ph↗
arXiv2000

Minimal Escape velocities

We give a new derivation of the minimal velocity estimates for unitary evolutions with some optimal bounds.

math-ph↗
arXiv1999

The Stability of Magnetic Vortices

We study the linearized stability of n-vortex solutions of the magnetic Ginzburg-Landau (or Abelian-Higgs) equations. We prove that the fundamental vortices (n=1,-1) are stable for all values of the coupling constant, k, and we prove that the higher-degree vortices (|n| > 1) are stable for k < 1 and unstable for k > 1. This resolves a long-standing conjecture.

math.AP↗