Adiabatic Decoupling and Time Evolution of Coherent States for Multi-component Systems

Abstract

Coherent states are well adapted to understand adiabatic decoupling for multi-components systems like the Dirac operator or systems obtained by reduction in the Born-Oppenheimer approximation [58]. Hagedorn [130] has done an accurate analysis of this using a variant of coherent states. The new fact for systems is that now states are vectors valued in \(\mathbb C^m\) for \(m\ge 2\) and the eigenvalues of the matrix of the system may cross so that one cannot diagonalize it smoothly. In the case when the eigenvalues are not crossing one can construct an adiabatic modification of the spectral projectors on the modes and compute the time evolution of coherent states following the same strategy as in the scalar case. Here we shall consider only non crossing cases where we can prove an adiabatic decoupling between the modes. For the more difficult crossing case we refer to the literature [130] where non-adiabatic transitions can be computed [101, 129] for more recent results.