The connection between the generator coordinate method and Bose expansions

It is shown that the generator coordinate method leads to a Bose expansion which in lowest order agrees with the RPA, and which for infinite order is equivalent to the Marumori expansion.     

A new boson expansion for odd-particle systems

The Beliaev-Zelevinsky boson expansion is extended to include systems with odd particle number. Closed forms, valid to all orders, are given for the particle-boson as well as boson-boson interaction.     

On the connection between the hyperspherical harmonic and generator coordinate methods for monopole vibrations

It is shown that, when the generating functions are Slater determinants of harmonic oscillator wave functions, the generator coordinate method for monopole vibrations is identical to the first approximation of the hyperspherical harmonic method.     

On calculating scattering phase shifts by the generator coordinate method

The generator coordinate equations for scattering phase shifts are solved for α-α scattering. Calculations are made in the coordinate representation and the results agree well with those obtained by a more laborious procedure which is formulated in the momentum representation. The results disagree with another calculation in the coordinate representation, where the scattering boundary condition is introduced less accurately.     

Variational principles for scattering in the generator coordinate method

The generator coordinate method (GCM) wave function is used as a trial function in a Kohn type variational principle for scattering phase shifts. It is shown that a GCM trial function is a solution of the variational equations if the Hill-Wheeler integral equation is satisfied subject to an appropriate boundary condition. A new method for introducing the scattering boundary condition is presented. There is a uniqueness theorem for the phase shift.     

Gamov states in the generator coordinate method

Starting from the generator coordinate theory, a method is developed for calculating-decay widths within a microscopic dynamical theory. Antisymmetrization is taken into account exactly between all nucleons of the decaying system. For illustration, the method is applied to the-decay of⁸Be and²⁰Ne.     

Pairing vibrational states and the generator coordinate method

The pairing vibrational states and the two-neutron transfer cross sections between these states are calculated in Ni, Sn and Pb isotopes by the generator coordinate method (GCM). The particle number fluctuation of the BCS functions is handled by projecting in a good approximation on sharp particle numbers. The results agree quite well with the experimental data.     

Gamov states in the generator coordinate method

Starting from the generator coordinate theory, a method is developed for calculatingα-decay widths within a microscopic dynamical theory. Antisymmetrization is taken into account exactly between all nucleons of the decaying system. For illustration, the method is applied to theα-decay of⁸Be and²⁰Ne.     

Collective modes in the generator coordinate method

Using the harmonic version of the generator coordinate method, and Skyrme interaction, the frequencies of the isoscalar breathing and quadrupole modes are related to the relevant incompressibility coefficients. The possibility of extending this to spin modes is also examined. It is found that a spin incompressibility coefficient is negative for a particular set of Skyrme parameter for⁴He. Other sets produce low positive values and these in turn could imply a relatively low lyingS=2,T=1 state. The replacement of the three-body term by the density-dependent one, suggested by Chang provides a cure for this pathology.     

Towards a macroscopic generator coordinate method

Collective quantities are defined as macroscopic statistical averages over many level crossing points where microscopic densities are redistributed. Accordingly, the generator coordinate method (GCM) is reconsidered. It is concluded that, contrary to earlier arguments, the macroscopically defined inertia parameter which appears in the GCM Hamiltonian has a finite value close to that obtained using traditional theories assuming the existence of the adiabatic BCS ground state.     

The generator coordinate method and the theory of metals

An application is made of the Generator Coordinate Method to a system of permions interacting with bosons. The problem treated in the present work is the collective dynamics of a metal regarded as an electron system interacting dynamically with the phonons of the lattice. A generalization of the Bohn-Staver formula for the sound velocity in metals is obtained.     

A mathematical foundation for discretisation techniques in the generator coordinate method

A different presentation of the Generator Coordinate Method is used to explain why, in spite of their being bad approximations to the Hill-Wheeler integral, discretisation techniques are so successful in GCM. A particular technique, which then appears appropriate, is discussed in detail.     

The coulomb problem for scattering states in the generator coordinate method

The proportionality of the Hill-Wheeler amplitude ? and of the scattering wave function g for large separation distance is proved for Coulomb + nuclear scattering. Practical problems are discussed.     

Effective nucleus-nucleus potentials derived from the generator coordinate method

The equivalence of the generator coordinate method (GCM) and the resonating group method (RGM) and the formal equivalence of the RGM and the orthogonality condition model (OCM) lead to a relation connecting the effective nucleus-nucleus potentials of the OCM with matrix elements of the GCM. This relation may be used to derive effective nucleus-nucleus potentials directly from GCM matrix elements without explicit reference to the potentials of the RGM. In a first application local and l-independent effective potentials are derived from diagonal GCM matrix elements which represent the energy surfaces of a two-centre shell model. Using these potentials the OCM can reproduce the results of a full RGM calculation very well for the elastic scattering of two α-particles and fairly well for elastic ¹⁶O-¹⁶O scattering.     

On the analytical unfolding of generator coordinate kernels for unequal fragments

The generator coordinate formalism breaks down for scattering and reactions involving fragments of unequal size. Analytical expressions are given through which, even in that case, the resonating group kernels can be extracted from those of the generator coordinate method. The possibility of defining a generator coordinate density matrix is discussed.     

Derivation of collective potential and mass from the generator coordinate method

The Hill-Wheeler equation of the generator coordinate method is approximated by a local collective Schrödinger equation. General expressions for the potential and the mass parameter are obtained by a symmetrized moment expansion. The validity of the approximation is tested for several examples where the exact solution is known. These include the Gaussian overlap with harmonic and anharmonic interaction, the Lipkin model, and monopole resonances of spherical light nuclei. In all cases, surprisingly close agreement with the exact solution is found. Other possible applications of the formalism are indicated.