97 research outputs found
The Flux-Across-Surfaces Theorem for a Point Interaction Hamiltonian
The flux-across-surfaces theorem establishes a fundamental relation in
quantum scattering theory between the asymptotic outgoing state and a quantity
which is directly measured in experiments. We prove it for a hamiltonian with a
point interaction, using the explicit expression for the propagator. The proof
requires only assuptions on the initial state and it covers also the case of
zero-energy resonance. We also outline a different approach based on
generalized eigenfunctions, in view of a possible extension of the result.Comment: AMS-Latex file, 11 page
Gell-Mann and Low formula for degenerate unperturbed states
The Gell-Mann and Low switching allows to transform eigenstates of an
unperturbed Hamiltonian into eigenstates of the modified Hamiltonian . This switching can be performed when the initial eigenstate is not
degenerate, under some gap conditions with the remainder of the spectrum. We
show here how to extend this approach to the case when the ground state of the
unperturbed Hamiltonian is degenerate. More precisely, we prove that the
switching procedure can still be performed when the initial states are
eigenstates of the finite rank self-adjoint operator \cP_0 V \cP_0, where
\cP_0 is the projection onto a degenerate eigenspace of
Spectral and scattering theory for some abstract QFT Hamiltonians
We introduce an abstract class of bosonic QFT Hamiltonians and study their
spectral and scattering theories. These Hamiltonians are of the form
H=\d\G(\omega)+ V acting on the bosonic Fock space \G(\ch), where
is a massive one-particle Hamiltonian acting on and is a Wick
polynomial \Wick(w) for a kernel satisfying some decay properties at
infinity. We describe the essential spectrum of , prove a Mourre estimate
outside a set of thresholds and prove the existence of asymptotic fields. Our
main result is the {\em asymptotic completeness} of the scattering theory,
which means that the CCR representations given by the asymptotic fields are of
Fock type, with the asymptotic vacua equal to the bound states of . As a
consequence is unitarily equivalent to a collection of second quantized
Hamiltonians
Bloch bundles, Marzari-Vanderbilt functional and maximally localized Wannier functions
We consider a periodic Schroedinger operator and the composite Wannier
functions corresponding to a relevant family of its Bloch bands, separated by a
gap from the rest of the spectrum. We study the associated localization
functional introduced by Marzari and Vanderbilt, and we prove some results
about the existence and exponential localization of its minimizers, in
dimension d < 4. The proof exploits ideas and methods from the theory of
harmonic maps between Riemannian manifolds.Comment: 37 pages, no figures. V2: the appendix has been completely rewritten.
V3: final version, to appear in Commun. Math. Physic
Infrared problem for the Nelson model on static space-times
We consider the Nelson model with variable coefficients and investigate the
problem of existence of a ground state and the removal of the ultraviolet
cutoff. Nelson models with variable coefficients arise when one replaces in the
usual Nelson model the flat Minkowski metric by a static metric, allowing also
the boson mass to depend on position. A physical example is obtained by
quantizing the Klein-Gordon equation on a static space-time coupled with a
non-relativistic particle. We investigate the existence of a ground state of
the Hamiltonian in the presence of the infrared problem, i.e. assuming that the
boson mass tends to 0 at infinity
Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential
Coupled-mode systems are used in physical literature to simplify the
nonlinear Maxwell and Gross-Pitaevskii equations with a small periodic
potential and to approximate localized solutions called gap solitons by
analytical expressions involving hyperbolic functions. We justify the use of
the one-dimensional stationary coupled-mode system for a relevant elliptic
problem by employing the method of Lyapunov--Schmidt reductions in Fourier
space. In particular, existence of periodic/anti-periodic and decaying
solutions is proved and the error terms are controlled in suitable norms. The
use of multi-dimensional stationary coupled-mode systems is justified for
analysis of bifurcations of periodic/anti-periodic solutions in a small
multi-dimensional periodic potential.Comment: 18 pages, no figure
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
Adiabatically coupled systems and fractional monodromy
We present a 1-parameter family of systems with fractional monodromy and
adiabatic separation of motion. We relate the presence of monodromy to a
redistribution of states both in the quantum and semi-quantum spectrum. We show
how the fractional monodromy arises from the non diagonal action of the
dynamical symmetry of the system and manifests itself as a generic property of
an important subclass of adiabatically coupled systems
Quasiperiodic functions theory and the superlattice potentials for a two-dimensional electron gas
We consider Novikov problem of the classification of level curves of
quasiperiodic functions on the plane and its connection with the conductivity
of two-dimensional electron gas in the presence of both orthogonal magnetic
field and the superlattice potentials of special type. We show that the
modulation techniques used in the recent papers on the 2D heterostructures
permit to obtain the general quasiperiodic potentials for 2D electron gas and
consider the asymptotic limit of conductivity when . Using the
theory of quasiperiodic functions we introduce here the topological
characteristics of such potentials observable in the conductivity. The
corresponding characteristics are the direct analog of the "topological
numbers" introduced previously in the conductivity of normal metals.Comment: Revtex, 16 pages, 12 figure
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