Adiabatic theorems for dense point spectra
The time dependent Schrödinger equation in Hilbert space is considered for slow variations of the Hamiltonian. Adiabatic theorems are proven for two classes of operators: perturbations of Hamiltonians with dense point spectrum, and Hamiltonians with dense point spectrum whose eigenvalues and -vectors depend smoothly on time. The usual gap condition on the spectrum of the operator [see \textit{J. E. Avron}, \textit{R. Seiler} and \textit{L. G. Yaffe}, Commun. Math. Phys. 110, 33-49 (1987; Zbl 0626.58033)] is replaced in the first case by the requirement that the perturbation does not couple the eigenspaces to much; in the second case the eigenvectors are supposed to stay localized.
scientific article; zbMATH DE number 4169270