Real stabilization of resonance states employing two parameters: basis set size and coordinate scaling

The resonance states of one- and two-particle Hamiltonians are studied using variational expansions with real basis-set functions. The resonance energies, $E_r$, and widths, $\mathrm{\Gamma }$, are calculated using the density of states and an ${\mathcal{L}}^2$ golden rule-like formula. We present a recipe to select adequately some solutions of the variational problem. The set of approximate energies obtained shows a very regular behaviour with the basis-set size, $N$. Indeed, these particular variational eigenvalues show a quite simple scaling behaviour and convergence when $N\to \mathrm{\infty }$. Following the same prescription to choose particular solutions of the variational problem we obtain a set of approximate widths. Using the scaling function that characterizes the behaviour of the approximate energies as a guide, it is possible to find a very good approximation to the actual value of the resonance width.