Twisted Pseudodifferential Calculus and Application to the Quantum Evolution of Molecules

TL;DR

This paper develops a pseudodifferential calculus with operator-valued symbols to analyze the quantum evolution of molecules, especially under Coulomb interactions, providing explicit symbol computations and long-time wave packet propagation insights.

Contribution

It introduces an abstract pseudodifferential calculus tailored for Coulomb interactions and applies it to molecular quantum dynamics within the Born-Oppenheimer approximation.

Findings

Reduction of molecular evolution to semiclassical operators with explicit symbols

Analysis of wave packet propagation up to Ehrenfest time scales

Framework applicable to molecules with spectral gaps in electronic Hamiltonians

Abstract

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.)