Structural analysis of a collective subspace in the dynamical generator coordinate method

TL;DR

This paper analyzes the structure of the collective subspace in the dynamical generator coordinate method (DGCM), revealing it can be expressed as a simple tensor product under certain conditions, aiding understanding of collective motions in nuclear theory.

Contribution

It provides a detailed analytical characterization of the collective subspace in DGCM, showing it can be simplified to a tensor product form under feasible conditions, applicable to various methods.

Findings

The collective subspace in DGCM is restricted to a specific tensor product and direct sum form.

Under certain conditions, the subspace can be expressed as a simple tensor product of collective and non-collective parts.

The analysis applies to the variation after projection method, revealing an untwisted structure of the function space.

Abstract

In nuclear theory, the generator coordinate method (GCM), a type of configuration mixing method, is often used for the microscopic description of collective motions. However, the GCM has a problem that a structure of the collective subspace, which is the Hilbert space spanned by the configurations, is not generally understood. In this paper, I investigate the structure of the collective subspace in the dynamical GCM (DGCM), an improved version of the GCM. I then show that it is restricted to a specific form that combines tensor products and direct sums under reasonable conditions. By imposing additional specific conditions that are feasible in actual numerical calculations, it is possible to write the collective subspace as a simple tensor product of the collective part and the others. These discussions are not dependent on the details of the function space used for generating the configurations and can be applied to various methods, including the mean-field theory. Moreover, this analytical technique can also be applied to a variation after projection method (VAP), then which reveals that under a specific condition, the function space of the VAP has an untwisted structure. These consequences can provide powerful tools for discussing the collective motions with the DGCM or the GCM.