Gaussian overlap approximation

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Gaussian overlap approximation

To compare cranking expressions with those obtained within the GOA, it is convenient to introduce the $\cal S$ matrices [23]:

$\begin{displaymath} {\cal S}_{ij}^{(K)}=\sum_{\mu,\nu} \frac{\langle\mu \vert ... ...gle} {(\breve{E}_\mu+\breve{E}_\nu)^{K}} (\eta^+_{\mu\nu})^2. \end{displaymath}$

(61)

It is immediately seen that for the mass tensor of Eq. (60) one has ${\cal M}^C = {\cal S}^{(3)}$ . In the case of GOA, also assuming weak state dependence of pairing and neglecting the derivatives of $\lambda$ and $\Delta_{\mu\bar\mu}$ , one obtains [24,23]

$\begin{displaymath} {\cal M}^{\rm GOA}= {\cal S}^{(2)}\left[{\cal S}^{(1)}\right]^{-1}{\cal S}^{(2)}. \end{displaymath}$

(62)

Evaluating the matrix elements of

entering Eq. (61) perturbatively, one can express ${\cal S}$ explicitly through the matrix elements of the constraining field operators $\hat Q_i$ :

$\begin{displaymath} {\cal S}^{(K)}=\frac{1}{4}\left[{ M}^{(1)}\right]^{-1}{ M}^{(K)} \left[{ M}^{(1)}\right]^{-1}, \end{displaymath}$

(63)

where the energy-weighted moments ${M}^{(K)}$ are given by

$\begin{displaymath} M^{(K)}_{ij} = \sum_{\mu\nu} \frac{\langle\mu\vert\hat Q... ...gle} {(\breve{E}_\mu+\breve{E}_{\nu})^K} (\eta_{\mu\nu}^+)^2. \end{displaymath}$

(64)

Jacek Dobaczewski 2010-07-28