Exponential approach to the adiabatic limit and the Landau-Zener formula. (English) Zbl 0769.34006

Jakšić, V.; Segert, J.

Rev. Math. Phys. 4, No. 4, 529-574 (1992).

Summary: We study the adiabatic limit for Hamiltonians with certain complex- analytic dependence on the time variable. We show that the transition probability from a spectral band that is separated by gaps is exponentially small in the adiabatic parameter. We find sufficient conditions for the Landau-Zener formula, and its generalization to nondiscrete spectrum, to bound the transition probability.

MSC:

34M99	Ordinary differential equations in the complex domain
30D55	\(H^p\)-classes (MSC2000)
45L05	Theoretical approximation of solutions to integral equations
81Q99	General mathematical topics and methods in quantum theory

Keywords:

adiabatic limit; Hamiltonians; complex-analytic dependence on the time variable; transition probability; Landau-Zener formula; nondiscrete spectrum