The generator-coordinate theory as a flexible formulation of the many-body Schrödinger equation

Abstract

By using generator coordinates as state labels we formulate the generator-coordinate theory of collective motion as non-orthogonal representations of the many-body Schrödinger equation in a subspace of the Hilbert space of many-body state vectors. The weight function of the usual generator-coordinate theory is generalized to become the components of the state vector in one of the two bi-orthogonal representations labelled by generator coordinates. The well-known case of the Gaussian-overlap approximation is studied in order to show how the new formalism also permits a solution of the problem using the original real generator coordinates. The concept of generator coordinates is clarified by studying (i) the connection between the redundancy of generator coordinates and the linear dependence of base vectors labelled by these generator coordinates, and (ii) the construction of states having the required properties under translation and rotation. Finally, the consideration of rotational properties leads to a double-projection method for constructing internal states of good angular momentum and for removing spurious states of c.m. motion.