A new look at Mourre's commutator theory

Mourre's commutator theory is a powerful tool to study the continuous spectrum of self-adjoint operators and to develop scattering theory. We propose a new approach of its main result, namely the derivation of the limiting absorption principle from a so called Mourre estimate. We provide a new interpretation of this result

Condition de non-capture pour des opérateurs de Schrödinger semi-classiques à potentiels matriciels.

An erratum is available here and published in MPEJ, vol. 13, 2007.International audienceWe consider semi-classical Schrödinger operators with matrix-valued, long-range, smooth potential, for which different eigenvalues may cross on a codimension one submanifold. We denote by $h$ the semiclassical parameter and we consider energies above the bottom of the essential spectrum. Under some invariance condition on the matricial structure of the potential near the eigenvalues crossing and some structure condition at infinity, we prove that the boundary values of the resolvent at energy $\lambda$, as bounded operators on suitable weighted spaces, are $O(h^{-1})$ if and only if $\lambda$ is a non-trapping energy for all the Hamilton flows generated by the eigenvalues of the operator's symbol.On considère des opérateurs de Schrödinger semi-classiques à potentiels matriciels, lisses et de longue portée, dont différentes valeurs propres peuvent se croiser sur une sous-variété de codimension une. On note par h le paramètre semi-classique et on s’intéresse aux énergies strictement supérieures au bas du spectre essentiel. Sous une certaine condition d’invariance le long du croisement, portant sur la structure matricielle du potentiel, et sous une certaine condition de structure à l’infini, on démontre que les valeurs aux bords de la résolvante à l’énergie λ, vues commeopérateurs bornés sur des espaces à poids convenables, sont O(h−1) si et seulement si λ est une énergie non-captive pour le flot hamiltonien générée par chaque valeur propre du symbole de l’opérateur

On the mathematical treatment of the Born-Oppenheimer approximation

Motivated by a paper by B.T. Sutcliffe and R.G. Woolley, we present the main ideas used by mathematicians to show the accuracy of the Born-Oppenheimer approximation for molecules. Based on mathematical works on this approximation for molecular bound states, in scattering theory, in resonance theory, and for short time evolution, we give an overview of some rigourous results obtained up to now. We also point out the main difficulties mathematicians are trying to overcome and speculate on further developments. The mathematical approach does not fit exactly to the common use of the approximation in Physics and Chemistry. We criticize the latter and comment on the differences, contributing in this way to the discussion on the Born-Oppenheimer approximation initiated by B.T. Sutcliffe and R.G. Woolley. The paper neither contains mathematical statements nor proofs. Instead we try to make accessible mathematically rigourous results on the subject to researchers in Quantum Chemistry or Physics

Semiclassical resolvent estimates for Schroedinger operators with Coulomb singularities

Consider the Schroedinger operator with semiclassical parameter h, in the limit where h goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator's resolvent at a positive energy E are bounded by O(1/h) if and only if the associated Hamilton flow is non-trapping at energy E. In the present paper, we extend this result to the case where the potential may possess Coulomb singularities. Since the Hamilton flow then is not complete in general, our analysis requires the use of an appropriate regularization.Comment: 39 pages, no figures, corrected versio

On Factorization of Molecular Wavefunctions

Recently there has been a renewed interest in the chemical physics literature of factorization of the position representation eigenfunctions \{$\Phi$\} of the molecular Schr\"odinger equation as originally proposed by Hunter in the 1970s. The idea is to represent $\Phi$ in the form $\varphi\chi$ where $\chi$ is \textit{purely} a function of the nuclear coordinates, while $\varphi$ must depend on both electron and nuclear position variables in the problem. This is a generalization of the approximate factorization originally proposed by Born and Oppenheimer, the hope being that an `exact' representation of $\Phi$ can be achieved in this form with $\varphi$ and $\chi$ interpretable as `electronic' and `nuclear' wavefunctions respectively. We offer a mathematical analysis of these proposals that identifies ambiguities stemming mainly from the singularities in the Coulomb potential energy.Comment: Manuscript submitted to Journal of Physics A: Mathematical and Theoretical, May 2015. Accepted for Publication August 24 201

Limited regularity of a specific electronic reduced density matrix for molecules

We consider an electronic bound state of the usual, non-relativistic, molecular Hamiltonian with Coulomb interactions, fixed nuclei, and N electrons (N>1). Near appropriate electronic collisions, we prove that the (N-1)-particle electronic reduced density matrix is not smooth.Comment: 33 page

Besov estimates in the high-frequency Helmholtz equation, for a non-trapping and C2 potential

AbstractWe study the high-frequency Helmholtz equation, for a potential having C2 smoothness, and satisfying the non-trapping condition. We prove optimal Morrey–Campanato estimates that are both homogeneous in space and uniform in the frequency parameter. The homogeneity of the obtained bounds, together with the weak assumptions we require on the potential, constitute the main new result in the present text. Our result extends previous bounds obtained by Perthame and Vega, in that we do not assume the potential satisfies the virial condition, a strong form of non-trapping