Relation of $a^{†}a$ terms to higher-order terms in the adiabatic expansion for large-amplitude collective motion

Abstract

We investigate the relation of $a^{†}a$ terms in the collective operator to the higher-order terms in the adiabatic self-consistent collective coordinate (ASCC) method. In the ASCC method, a state vector is written as $e^{i\hat{G}(q,p,n)}|\varphi (q)⟩$ with $\hat{G}(q,p,n)$ which is a function of collective coordinate $q$, its conjugate momentum $p$ and the particle number $n$. According to the generalized Thouless theorem, $\hat{G}$ can be written as a linear combination of two-quasiparticle creation and annihilation operators $a_{\mu }^{†}{a}_{\nu }^{†}$ and $a_{\nu }{a}_{\mu }$. We show that, if $a^{†}a$ terms are included in $\hat{G}(q,p,n)$, it corresponds to the higher-order terms in the adiabatic expansion of $\hat{G}$. This relation serves as a prescription to determine the higher-order collective operators from the $a^{†}a$ part of the collective operator, once it is given without solving the higher-order equations of motion.

Comments: 19 pages, no figure