On the adiabatic theorem in quantum theory

Vladimir Nikolaevich Likhachev, Yu. S. Tyupkin, A. S. Shvart︠s︡

Sven Bachmann, Wojciech De Roeck, Martin Fraas

Tosio Kato

Cao Huaixi

Quantum adiabatic theorem(QAT)is one of the important principles in quantum theory.With the discovery of the inconsistency of the QAT in application,it has received much more attention from physicists and mathematicians.Based on a brief review about the recent researches on QAT,some discussions on this topic are introduced,including the concepts of uniformly slowly evolving,CPT-frame,PT-symmetric operator,quantum adiabatic evolution theorem,quantum adiabatic approximation theorem,and error estimation of adiabatic approximation.

S. Fishman, A. Soffer

We employ the recently developed multi-time scale averaging method to study the large time behavior of slowly changing (in time) Hamiltonians. We treat some known cases in a new way, such as the Zener problem, and we give another proof of the adiabatic theorem in the gapless case. We prove a new uniform ergodic theorem for slowly changing unitary operators. This theorem is then used to derive the adiabatic theorem, do the scattering theory for such Hamiltonians, and prove some classical propagation estimates and asymptotic completeness.

V. S. Buslaev, E. A. Grinina