Adiabatic theorem and generalized geometrical phase in the case of continuous spectra

By defining "a virtual gap" for the continuous spectrum through the notion of eigendifferential (Weyl's packet) and using the differential projector operator, we present a rigorous demonstration and discussion of the quantum adiabatic theorem for systems having a nondegenerate continuous spectrum. An explicit formula for a generalized geometrical phase is derived in terms of the eigenstates of the Hamiltonian. Examples are given for illustration.