An Adiabatic Theorem without a Gap Condition

Abstract

We prove the adiabatic theorem for quantum evolution without the traditional gap condition. All that this adiabatic theorem needs is a (piecewise) twice differentiable finite dimensional spectral projection. The result implies that the adiabatic theorem holds for the ground state of atoms in quantized radiation field. The general result we prove gives no information on the rate at which the adiabatic limit is approached. With additional spectral information one can also estimate this rate.

Adiabatic Evolution of Coupled Waves for a Schrödinger-Korteweg-de Vries System

“…The main ingredients of our analysis are generalizations of adiabatic theorems for bound and metastable quantum states to the case at hand, see [10], and also [11][12][13][14]16], and a straightforward extension of results on the effective dynamics of solitons for KdV equation over a slowly varying bottom to the random case, [7], see also [5,6,8,9] for relevant results.…”

Tosio Kato’s work on non-relativistic quantum mechanics: part 2

“…Nevertheless, there are results on adiabatic theorems without gaps, see Avron et al [29] and Hagedorn [215] for some special situations and Avron-Elgart [22] for a very general result. Teufel [641] has an alternate proof for this Avron-Elgart result and he has a book [642] on the subject.…”

Reflections on Friction in Quantum Mechanics

“…We emphasize that if level-crossing does occur, Allahverdyan and Nieuwenhuizen show that the quasistatic protocol may not be the optimal one. A quasistatic timescale for the adiabatic theorem can still be defined in this case [12,13].…”