 
Summary: Communications in
Commun. Math. Phys. 128, 497507 (1990) Mathematical
Physics
© SpringerVerlag1990
Adiabatic Theorems for Dense Point Spectra*
J. E. Avron**, J.S. Howland*** and B. Simon
Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA
Abstract. Weprove adiabatic theorems in situations where the Hamiltonian
has dense point spectrum. The gap condition of the standard adiabatic theorems
is replaced byan appropriate condition on the ineffectiveness ofresonances.
1. Introduction
The prototype Adiabatic Theorem in quantum mechanics asserts that for operators
with discrete spectra, in theadiabatic limit, thephysical evolution takes an
instantaneous eigenstate att = 0to the corresponding instantaneous eigenstateat
a later time [3,8]. More generally [1], with no assumptions about the nature of
the instantaneous spectrum, but provided it has a gap foralltimes, the physical
evolution in the adiabatic limit respects the splitting of the Hubert space into
spectral subspaces: Astate inthe subspace below the gap att = 0will evolve toa
state that lies in the corresponding subspace below the corresponding gap at time
t. While there arevarious kinds of adiabatic theorems that deal with the two
