Asymptotic behavior finite-size scaling

The relevant questions to be addressed at this point are related to the calculation of Xc, the determination of the leading term in the asymptotic behavior of E(k) near and the existence of a square-integrable eigenfunction at the threshold. In the forthcoming sections we will develop the finite-size scaling as a powerful method to obtain accurate numerical estimations of critical parameters. Here we will present some exact results. 

The convergence law of the results of the PR method is related to the corrections to the finite-size scaling. From Eq. (55) we expect that at the critical value of nuclear charge the correlation length is linear in N. In Fig. 9 we plot the correlation length of the finite pseudosystem (evaluated at the exact critical point Xc) as a function of the order N. The linear behavior shows that the asymptotic equation [Eq. (60)] for the correlation length holds for very low values of N [87]. 

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