Adiabatic evolution for systems with infinitely many eigenvalue crossings

We formulate an adiabatic theorem adapted to models that present an instant aneous eigenvalue experiencing an infinite number of crossings with the res t of the spectrum. We give an upper bound on the leading correction terms w ith respect to the adiabatic limit. The result requires only differentiabil ity of the considered projector, and some geometric hypothesis on the local behavior of the eigenvalues at the crossings. (C) 1999 American Institute of Physics. [S0022-2488(99)00511-3].