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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 $h^i$ 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