Generalized valley approximation applied to a schematic model of the monopole excitation

In recent years we have developed a new mathematical treatment of large amplitude collective motion in the adiabatic limit and formulated a successful approximation method, called the generalized valley approximation. In this paper we discuss its application to adiabatic time-dependent Hartree theory, for which the method is ideally suited. We apply the method first to an exactly solvable limiting case (the Suzuki model), for which we have shown in a previous paper that the usual form of adiabatic time-dependent Hartree theory is not general enough to yield the exact solution. We introduce an extended theory that remedies this deficiency. The modified theory has also been applied to monopole models close to the Suzuki model that are not exactly solvable. The algorithm developed for this case is sufficiently general to serve as a prototype for those necessary to study more complex realistic models of collective motion.