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

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

Exponentially accurate semiclassical asymptotics of low-lying eigenvalues for 2×2 matrix Schrödinger operators

AbstractWe consider a simple molecular-type quantum system in which the nuclei have one degree of freedom and the electrons have two levels. The Hamiltonian has the form H(ɛ)=−ɛ42∂2∂y2+h(y),where h(y) is a 2×2 real symmetric matrix. Near a local minimum of an electron level E(y) that is not at a level crossing, we construct quasimodes that are exponentially accurate in the square of the Born–Oppenheimer parameter ɛ by optimal truncation of the Rayleigh–Schrödinger series. That is, we construct Eɛ and Ψɛ, such that ‖Ψɛ‖=O(1) and ‖(H(ɛ)−Eɛ)Ψɛ‖<Λexp(−Γ/ɛ2), where Γ>0

Response Ranking with Deep Matching Networks and External Knowledge in Information-seeking Conversation Systems

Intelligent personal assistant systems with either text-based or voice-based conversational interfaces are becoming increasingly popular around the world. Retrieval-based conversation models have the advantages of returning fluent and informative responses. Most existing studies in this area are on open domain "chit-chat" conversations or task / transaction oriented conversations. More research is needed for information-seeking conversations. There is also a lack of modeling external knowledge beyond the dialog utterances among current conversational models. In this paper, we propose a learning framework on the top of deep neural matching networks that leverages external knowledge for response ranking in information-seeking conversation systems. We incorporate external knowledge into deep neural models with pseudo-relevance feedback and QA correspondence knowledge distillation. Extensive experiments with three information-seeking conversation data sets including both open benchmarks and commercial data show that, our methods outperform various baseline methods including several deep text matching models and the state-of-the-art method on response selection in multi-turn conversations. We also perform analysis over different response types, model variations and ranking examples. Our models and research findings provide new insights on how to utilize external knowledge with deep neural models for response selection and have implications for the design of the next generation of information-seeking conversation systems.Comment: Accepted by the 41th International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR 2018), Ann Arbor, Michigan, U.S.A. July 8-12, 2018 (Full Oral Paper

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

Analytic Wave Front Set for Solutions to Schro&#776;dinger Equations II \u2013 Long Range Perturbations

This paper is a continuation of a previous work, where short range perturbations of the flat Euclidian metric where considered. Here, we generalize the results to long-range perturbations (in particular, we can allow potentials subquadratic at infinity). More precisely, we construct a modified quantum free evolution acting on Sjo&#776;strand\u2019s spaces, and we characterize the analytic wave front set of the solution to the Schro&#776;dinger equation, in terms of the microlocal semiclassical exponential decay of the corresponding modified quantum free evolution. The result is valid for t &lt; 0 near the forward non trapping points, and for t > 0 near the backward non trapping points