...http://www-dft.ts.infn.it/ ansoldi¹

E-mail address: ansoldi@trieste.infn.it

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... formalism²

But see also [8] and references therein for a complementary approach which also tackles the issue of stability.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... shell³

To avoid any possible confusion, in what follows we are going to use square brackets only with this meaning, according to the following definition.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... dynamics⁴

We denote with an overbar quantities, let us say $g$, which are function of the rescaled radial coordinate $x$, although in many cases we have $\bar{g} (x) = g (R)$. For the sake of precision, note that $f _{\mathrm{in}} (R) = \bar{f} _{\mathrm{in}} (x)$, $f _{\mathrm{out}} (R) = \bar{f} _{\mathrm{out}} (x)$, $\sigma _{\mathrm{in}} (R) = \bar{\sigma} _{\mathrm{in}} (x)$, $\sigma _{\mathrm{out}} (R) = \bar{\sigma} _{\mathrm{out}} (x)$, and $V (R) = \bar{V} (x)$.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... trajectory⁵

Nevertheless it equals $x _{\mathrm{max}}$ for $\Theta = 1 / 2$. This is an interesting limiting situation in the case of tunnelling across the potential barrier, which will be discussed elsewhere.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... of⁶

We work in units where $l _{P} ^{2} = \hbar = c = G \equiv 1$.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... barrier⁷

This will be the topic of a forthcoming paper.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... properties⁸

We define ${\mathcal{F}}$, ${\mathcal{R}}$ and ${\mathcal{D}}$ according to the first two $\equiv$'s of the equation below.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.