8 research outputs found
On the Dynamics of solitons in the nonlinear Schroedinger equation
We study the behavior of the soliton solutions of the equation
i((\partial{\psi})/(\partialt))=-(1/(2m)){\Delta}{\psi}+(1/2)W_{{\epsilon}}'({\psi})+V(x){\psi}
where W_{{\epsilon}}' is a suitable nonlinear term which is singular for
{\epsilon}=0. We use the "strong" nonlinearity to obtain results on existence,
shape, stability and dynamics of the soliton. The main result of this paper
(Theorem 1) shows that for {\epsilon}\to0 the orbit of our soliton approaches
the orbit of a classical particle in a potential V(x).Comment: 29 page
Effective dynamics for particles coupled to a quantized scalar field
We consider a system of N non-relativistic spinless quantum particles
(``electrons'') interacting with a quantized scalar Bose field (whose
excitations we call ``photons''). We examine the case when the velocity v of
the electrons is small with respect to the one of the photons, denoted by c
(v/c= epsilon << 1). We show that dressed particle states exist (particles
surrounded by ``virtual photons''), which, up to terms of order (v/c)^3, follow
Hamiltonian dynamics. The effective N-particle Hamiltonian contains the kinetic
energies of the particles and Coulomb-like pair potentials at order (v/c)^0 and
the velocity dependent Darwin interaction and a mass renormalization at order
(v/c)^{2}. Beyond that order the effective dynamics are expected to be
dissipative.
The main mathematical tool we use is adiabatic perturbation theory. However,
in the present case there is no eigenvalue which is separated by a gap from the
rest of the spectrum, but its role is taken by the bottom of the absolutely
continuous spectrum, which is not an eigenvalue.
Nevertheless we construct approximate dressed electrons subspaces, which are
adiabatically invariant for the dynamics up to order (v/c)\sqrt{\ln
(v/c)^{-1}}. We also give an explicit expression for the non adiabatic
transitions corresponding to emission of free photons. For the radiated energy
we obtain the quantum analogue of the Larmor formula of classical
electrodynamics.Comment: 67 pages, 2 figures, version accepted for publication in
Communications in Mathematical Physic
Adiabatic theorems for generators of contracting evolutions
We develop an adiabatic theory for generators of contracting evolution on
Banach spaces. This provides a uniform framework for a host of adiabatic
theorems ranging from unitary quantum evolutions through quantum evolutions of
open systems generated by Lindbladians all the way to classically driven
stochastic systems. In all these cases the adiabatic evolution approximates, to
lowest order, the natural notion of parallel transport in the manifold of
instantaneous stationary states. The dynamics in the manifold of instantaneous
stationary states and transversal to it have distinct characteristics: The
former is irreversible and the latter is transient in a sense that we explain.
Both the gapped and gapless cases are considered. Some applications are
discussed.Comment: 31 pages, 4 figures, replaced by the version accepted for publication
in CM
Freezing of Energy of a Soliton in an External Potential
In this paper we study the dynamics of a soliton in the generalized NLS with a small external potential \u3f5V of Schwartz class. We prove that there exists an effective mechanical system describing the dynamics of the soliton and that, for any positive integer r, the energy of such a mechanical system is almost conserved up to times of order \u3f5 12r. In the rotational invariant case we deduce that the true orbit of the soliton remains close to the mechanical one up to times of order \u3f5 12r