Suppose that one has a continuous Hamiltonian with spin-orbit interaction, for example
$H=-\dfrac{\mathbf{p}^2}{2m} +\kappa({\boldsymbol\sigma}\times\mathbf{p}) + U(x)$
and want to approximate this Hamiltonian with a tight-binding discrete model. This can be done with the method of finite differences (e.g., Datta, Quantum Transport), using the substitution
$(\partial^2_x \psi)_{x=x_n} \rightarrow \frac1{a^2}[\psi(x_{n+1}-2\psi(x_n)+\psi(x_{n-1})]$
$(\partial_x \psi)_{x=x_n} \rightarrow \frac1{2a}[\psi(x_{n+1}-\psi(x_{n-1})]$
Suppose now that one has the opposite problem, i.e., one has a tight-binding Hamiltonian and want to obtain informations on the system in the continuous limit. Just as an example, consider the tight-binding Hamiltonian
$H=\sum_{i=1}^N\sum_{ss'} \delta_{ss'} [u(r_i)-\mu] a^\dagger_{is} a_{is} + \delta_{ss'} t a^\dagger_{is} a_{i+1,s} + \imath\alpha\sigma^y_{ss'} a^\dagger_{is} a_{i+1,s'} + \text{h.c.}$
where $s$ and $t$ are the spin indexes. This Hamiltonian is in the general case not analytically solvable, since the parameters and $u(r_i)$ can depend on the lattice sites (e.g., the system can contain one or more impurities).
However, for finite $N$ one can always, in principle, diagonalize the Hamiltonian numerically and calculate some relevant physical quantities, for example the gap. I am explicitly interested in cases where the tight-binding model can be solved only numerically.
Is there a general method to take the continuous limit $N\rightarrow\infty$ and $Na=\text{constant}$ of a general discrete tight-binding model? To avoid confusion, I stress that I am not interested to the case where the system is infinite, but rather on the limit case where the system is continuous, i.e., not discrete, but of finite size $L=Na$.
One idea is to calculate numerically the spectra and the relevant physical quantities for increasing $N$ and with a decreasing lattice parameter $a\rightarrow 0$ with the constraint $Na=\text{constant}$ and to deduce the asymptotic behavior. Since in the continuous limit $t=\hbar^2/(2m a^2)$ and $\alpha=\kappa/(2a)$ ($\kappa$ is the SO parameter), this is equivalent to take the limit $N\rightarrow\infty$ and taking $t\propto N^2$ and $\alpha\propto N$.
Is this approach correct? Is it correct to assume the limit $N\rightarrow\infty$ with $t\propto N^2$ and $\alpha=\propto N$ at each step of the calculation? Is there a reference or a book that discuss the discrete to continuous limit?
Many thanks in advance.