27 research outputs found
On the Born-Oppenheimer approximation of diatomic molecular resonances
We give a new reduction of a general diatomic molecular Hamiltonian, without
modifying it near the collision set of nuclei. The resulting effective
Hamiltonian is the sum of a smooth semiclassical pseudodifferential operator
(the semiclassical parameter being the inverse of the square-root of the
nuclear mass), and a semibounded operator localised in the elliptic region
corresponding to the nuclear collision set. We also study its behaviour on
exponential weights, and give several applications where molecular resonances
appear and can be well located.Comment: 22 page
Building End-To-End Dialogue Systems Using Generative Hierarchical Neural Network Models
We investigate the task of building open domain, conversational dialogue
systems based on large dialogue corpora using generative models. Generative
models produce system responses that are autonomously generated word-by-word,
opening up the possibility for realistic, flexible interactions. In support of
this goal, we extend the recently proposed hierarchical recurrent
encoder-decoder neural network to the dialogue domain, and demonstrate that
this model is competitive with state-of-the-art neural language models and
back-off n-gram models. We investigate the limitations of this and similar
approaches, and show how its performance can be improved by bootstrapping the
learning from a larger question-answer pair corpus and from pretrained word
embeddings.Comment: 8 pages with references; Published in AAAI 2016 (Special Track on
Cognitive Systems
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
Molecular scattering and Born\u2013Oppenheimer approximation
In this paper, we study the scattering wave operators for diatomic molecules by using the Born\u2013Oppenheimer approximation. Assuming that the ratio h2 between the electronic and nuclear masses is small, we construct adiabatic wave operators that, under some non-trapping conditions, approximate the two-cluster wave operators up to any power of the parameter h
Widths of highly excited resonances in multidimensional molecular predissociation
We investigate the simple resonances of a 2 by 2 matrix of n-dimensional semiclassical Schr\uf6dinger operators that interact through a first order differential operator. We assume that one of the two (analytic) potentials admits a well with non empty interior, while the other one is non trapping and creates a barrier between the well and infinity. Under a condition on the resonant state inside the well, we find an optimal lower bound on the width of the resonance. The method of proof relies on Carleman estimates, microlocal propagation of the microsupport, and a refined study of a non involutive double characteristic problem in the framework of Sj\uf6strand's analytic microlocal theory
Born-Oppenheimer Reduction of Quantum Evolution of Molecules
We construct an abstract pseudodifferential calculus with operator-valued symbol, adapted to the treatment of Coulomb-type interactions, and we apply it to study the quantum evolution of molecules in the Born-Oppenheimer approximation, in the case where the electronic Hamiltonian admits a local gap in its spectrum. In particular, we show that the molecular evolution can be reduced to the one of a system of smooth semiclassical operators, the symbol of which can be computed explicitely. In addition, we study the propagation of certain wave packets up to long time values of Ehrenfest order