Slowly changing potential problems in Quantum Mechanics: Adiabatic theorems, ergodic theorems, and scattering

Journal Article · Fri Jul 15 00:00:00 EDT 2016 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4954498· OSTI ID:22596650

Fishman, S., E-mail: [e-mail] ^[1]; Soffer, A., E-mail: [e-mail] ^[2]

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1. Technion, Haifa 32000 (Israel)
2. Mathematics Department, Rutgers University, Piscataway, NJ 08854 (United States)

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.

• OSTI ID: 22596650
• Journal Information: Journal of Mathematical Physics, Vol. 57, Issue 7; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ASYMPTOTIC SOLUTIONS
HAMILTONIANS
QUANTUM MECHANICS
SCATTERING