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

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