But how is this quantity related to the original problem of Anderson? How is it related to the localization/delocalization of the eigenstates ?
Let say we have a finite disorder system of size $L$ and dimension $d$ in which we put some particule. If the system is closed, in the sense that the particle cannot get out of the system, then the possible states are bound to the system, such that the energy eigenstates form a discrete spectrum. Considering all these discrete eigenstates, one can compute the mean level spacing $\Delta E$ separating each eigenstate from each other, which is associated to the mean density of states $\nu$ (by unit volume) :
$$
\nu=\frac{1}{\Delta E\;L^d}.\tag{1}
$$
In turn, if the system is open, the particle can eventually get out of the system by reaching the boundary. The time $\tau_D$ it takes to the particle to diffuse to the boundary is determined such that :
$$
L\sim\sqrt{D\,\tau_D}\quad\text{i.e.}\quad\tau_D\sim\frac{L^2}{D},\tag{2}
$$
where $D$ is the diffusion coefficient.
This means that, through its diffusion in the disorder, the particle can only resolve individual eigenstates with an energy resolution $\delta E$ such that :
$$
\delta E\,\tau_D\sim\hbar\quad\text{i.e.}\quad \delta E\sim\frac{\hbar D}{L^2}.\tag{3}
$$
where $\delta E$ is oftenly called the "Thouless energy".
Then, the Thouless criterion for localization [1] tells us that the system is Anderson localized when this energy resolution $\delta E$ is much smaller than the mean level spacing $\Delta E$. Indeed, in this limiting case $\delta E\ll\Delta E$, the particle can resolve each one of the eigenstate and occupy one of them for ever.
Conversly, in the limit where $\delta E\gg\Delta E$ the different energy levels overlap on each other, meaning that the particle can be coupled and jump from one state to another, thus leading to diffusion toward the boundary of the system and delocalization.
After this discussion, it seems quite natural to define the quantity :
$$
g=\frac{\delta E}{\Delta E}\sim\hbar D\,\nu\,L^{d-2}\tag{4}
$$
as an order parameter of the Anderson metal/insulator transition, for which $g\gg 1$ corresponds to the delocalized/metallic phase, and $g\ll 1$ to the localized/insulator phase.
At this point, one could argue that the link between this new quantity $g$ and the usual conductivity $G=\sigma\,S/L\sim\sigma\,L^{d-2}$ according to the Ohm's law is not very clear.
But since the conductivity $\sigma$ and the diffusion coefficient $D$ are linked one to each other through the Einstein relation :
$$
\sigma\propto e^2\,\nu\,D\quad\text{such that}\quad G\sim e^2\,\nu\,D\,L^{d-2}\tag{5}
$$
it is actually easy to see by comparing the expression (4) and (5) that $g$ has no dimension and gives the conductance $G\sim (e^2/h)\,g$ in unit of conductance quantum $e^2/h$.
So, to the questions :
Is g(L) determined by the Hamiltonian above ?
It is determined by the spectral properties of the eigenstates of the hamiltonian such as the density of states $\nu$ and the mean level spacing according to the size $L$ of the system.
or some more parameters are needed, say, the temperature ?
Everything that has been told before was for a system at $T=0$. From the point of view of the phase transition between localized and diffusive states, the only relevant information is the scaling of $g$ according to the size $L$ of system, which is completely determined by the $\beta$-function of the very famous scaling theory of localization.
For those who are interested in getting into the field of Anderson localization, I would advice to check the lecture notes of Les Houches School of Physics on "Ultracold gases and quantum information" (2009) given by Delande and Müller. IMO it is one of the rare references that achieve to introduce localization phenomena in simple terms while kipping tracks with the experiments on the subject.
[1] : a good review about this Thouless criterion can be found in Disordered electronic systems, P. A. Lee and T. V. Ramakrishnan, Rev.Mod.Phys. 57, 287.
