 
Summary: Commun. Math. Phys. 203, 445 463 (1999) Communications in
Mathematical
Physics
© SpringerVerlag 1999
Adiabatic Theorem without a Gap Condition
Joseph E. Avron, Alexander Elgart
Department of Physics, Technion, 32000 Haifa, Israel. Email: avron@physics.technion.ac.il
Received: 3 December 1998/ Accepted: 7 December 1998
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.
1. Introduction and Motivation
The adiabatic theorem of Quantum Mechanics describes the long time behavior of solu
tions of an initial value problem where the Hamiltonian generating the evolution depends
slowly on time. The theorem relates these solutions to spectral information of the in
stantaneous Hamiltonian.
Traditionally, the adiabatic theorem is stated for Hamiltonians that have an eigenvalue
