Exponential estimates in adiabatic quantum evolution

Abstract We review recent results concerning the exponential behaviour of transition probabilitiesacross a gap in the adiabatic limit of the time-dependent Schro¨dinger equation. They range froman exponential estimate in quite general situations to asymptotic Landau-Zener type formulaefor finite dimensional systems, or systems reducible to this case. 1 IntroductionThe notion of adiabatic evolution or adiabatic process is an important theoreticalconcept, which occurs at several places in Physics. In Quantum Mechanics, this processis usually described by the equation i¯h ∂∂t ′ ψ(t ′ ) = H(et ′ )ψ(t ′ ), where eis a small pa-rameter, such that 1/egives the typical time-scale over which the Hamiltonian changessignificantly. Setting ¯h= 1 and introducing a rescaled time, t= et ′ , we can rewrite thetime-dependent Schr¨odinger equation for the evolution operator Uasie∂∂tU(t,s) = H(t)U(t,s), U(s,s) = I , ∀t,s∈ R. (1)The adiabatic limit corresponds to the singular limit e→ 0 of the equation (1). If thestate of the system is an eigenfunction ψ(t