The scattering operator and the adiabatic theorem

Gautam Kumar Ghosh

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

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.

Pierre Lochak, Claude Meunier

Kyu Hwang Yeon, Jeong Ryeol Choi, Shou Zhang +1 more

The unitary operator corresponding to the classical canonical transformation that connects a general closed system to an open system under adiabatic conditions is found. The quantum invariant operator of the adiabatic open system is derived from the unitary transformation of the quantum Hamiltonian of the closed system. On the basis of these results, we investigate the evolution of the general quantum adiabatic system and construct a revised adiabatic theorem. The adiabatic theorem developed here exactly reduces to the well-known Berry adiabatic theorem when the control parameter