An Egorov Theorem for avoided crossings of eigenvalue surfaces

Abstract

We study nuclear propagation through avoided crossings of electron energy levels. We construct a surface hopping semigroup, which gives an Egorov-type description of the dynamics. The underlying time-dependent Schroedinger equation has a two-by-two matrix-valued potential, whose eigenvalue surfaces have an avoided crossing. Using microlocal normal forms reminiscent of the Landau-Zener problem, we prove convergence to the true solution in the semi-classical limit.

arXiv comment44 pages

References in corpus (5)

• Nonadiabatic nuclear dynamics of the ammonia cation studied by surface hopping classical trajectory calculations
• Adiabatic Decoupling and Time-Dependent Born-Oppenheimer Theory
• Superadiabatic transition histories in quantum molecular dynamics
• Determination of Non-Adiabatic Scattering Wave Functions in a Born-Oppenheimer Model
• A kinetic model for the transport of electrons in a graphene layer

Cited by in corpus (8)

• Computing quantum dynamics in the semiclassical regime
• Wavepackets in inhomogeneous periodic media: propagation through a one-dimensional band crossing
• A kinetic model for the transport of electrons in a graphene layer
• Observable Error Bounds of the Time-splitting Scheme for Quantum-Classical Molecular Dynamics
• Coherent electronic transport in periodic crystals
• Propagation of Wave Packets for Systems Presenting Codimension 1 Crossings
• Propagation of Coherent States through Conical Intersections
• Landau-Zener Formula in a "Non-Adiabatic" regime for avoided crossings