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Andre' Martinez

Publications and source records attributed to Andre' Martinez.

4 recordsLinked to original sources
arXiv2008

Twisted Pseudodifferential Calculus and Application to the 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. (This work has been accepted for publication as part of the Memoirs of the American Mathematical Society and will be published in a future volume.)

math.AP↗
arXiv2008

Analytic Wave Front Set for Solutions to Schrödinger Equations II -- Long Range Perturbations

This paper is a continuation of a paper by the authors: arXiv:0706.0415, where short range perturbations of the flat Euclidian metric where considered. Here, we generalize the results of the paper to long-range perturbations (in particular, we can allow potentials growing like $ ^{2-\varepsilon}$ at infinity). More precisely, we construct a modified quantum free evolution $G_0(-s, hD_z)$ acting on Sjöstrand's spaces, and we characterize the analytic wave front set of the solution $e^{-itH}u_0$ of the Schrödinger equation, in terms of the semiclassical exponential decay of $G_0(-th^{-1}, hD_z)T u_0$, where $T$ stands for the Bargmann-transform. The result is valid for $t<0$ near the forward non trapping points, and for $t>0$ near the backward non trapping points. It is an extension of a paper by Nakamura (arXiv:math/0605742) to the analytic framework.

math.AP↗
arXiv2007

Analytic wave front set for solutions to Schroedinger equation

This paper is a continuation of a previous paper by the same authors, where an analytic smoothing effect was proved for long-range type perturbations of the Laplacian $H_0$ on $\re^n$. In this paper, we consider short-range type perturbations $H$ of the Laplacian on $\re^n$, and we characterize the analytic wave front set of the solution to the Schrödinger equation: $e^{-itH}f$, in terms of that of the free solution: $e^{-itH_0}f$, for $t<0$ in the forward nontrapping region. The same result holds for $t>0$ in the backward nontrapping region. This result is an analytic analogue of results by Hassel and Wunsch and Nakamura.

math.AP↗