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
[1]
S. Krein,et al.
Linear Differential Equations in Banach Space
,
1972
.
[2]
D. Ter Haar,et al.
Collected Papers of L. D. Landau
,
1965
.
[3]
A. Lenard,et al.
Adiabatic invariance to all orders
,
1959
.
[4]
F. Wilczek,et al.
Geometric Phases in Physics
,
1989
.
[5]
Mikhail Vasilʹevich Fedori︠u︡k.
Méthodes asymptotiques pour les équations différentielles ordinaires linéaires
,
1987
.
[6]
N. Fröman,et al.
JWKB approximation : contributions to the theory
,
1965
.