second order (or smaller) in the rest of the matrix.

The matrix elements of $\mathcal{G}$, approximated to the most relevant order, are shown in the Table \[T:1\]. The determinant of this matrix turns out to be $$\begin{aligned}
G &\propto & (\beta_2^2 + 2 \beta_3^2)^2\ f^4(\beta_2, \beta_3) 
\ u_0^2 \ v^2 u^2 w^2\nonumber \\[2pt]
&\equiv & G_0(\beta_2,\beta_3)\ u_0^2 \ v^2 u^2 w^2 .
\label{E:2.7}\end{aligned}$$ The Pauli recipe for the quantization of the classical kinetic energy gives the Schrödinger equation $$\sum_{\mu \nu} \frac{1}{g} 
\frac{\partial}{\partial \xi_\mu} \left[ g 
\!\left( \mathcal{G}^{-1} \right) _{\mu \nu} 
\frac{\partial \Psi}{\partial \xi_\nu }
\right]+ \frac{2}{\hbar^2} \left[ E\!-\!V(\xi ) \right] \Psi\! =\!0
\label{E:a1.2}$$ where $g^2=G={\rm Det}\ \mathcal{G}$ and $\xi$ stays for the ensemble of the variables $\xi_\kappa$.

To our present purpose, this general treatment must be specialized, ([*e.g.*]{} with a proper choice of the arbitrary function $f(\beta_2,\beta_3)$ in the Eq.s \[E:2.3\]), keeping in mind a necessary condition: the Schrödinger equation for the quadrupole amplitude, when the octupole amplitude is constrained to small values by a proper restoring potential, must converge to that of Bohr and therefore, at the critical–point, to that of the X(5) model[^2]. As we shall see in the Section \[S:4.1\], this result is obtained with the choice 