# Green's functions of disordered tight binding models

A research problem has led me to calculate a Green's function of a tight binding model with both onsite disorder and hopping amplitudes which vary in space. Since so much is known about tight binding models I thought I'd ask if this or other similar problems have been encountered before, since it doesn't make much sense to re-invent the wheel.

More concretely: I have in mind a finite tight binding model with open boundary conditions, with a Hamiltonian given by

$$H = \sum_{j=0}^{N}\left(\epsilon_{j}|j\rangle\langle j| - t_{j} |j+1\rangle\langle j| -t_{j} |j\rangle\langle j+1| \right)$$

To restrict to $$N$$ sites, we set $$t_{N}=0$$. In a particular problem, we can determine the potentials $$\epsilon_{j}$$ and hopping amplitudes $$t_{j}$$, but this is not really the point; all that matters is that these parameters don't have any symmetries. However, we can assume that the hopping amplitudes vary slowly for sufficiently large $$j$$, and in this regime the potentials also vary slowly. This should correspond to something like a gradient expansion in a suitable continuum limit.

What I would now like to calculate is the Green's function defined by

$$H|G\rangle = |0\rangle$$

which is to say, the vector $$|G\rangle$$ which, when acted on by the Hamiltonian, projects onto the first tight binding site. I imagine this can be converted into a continuum differential equation which can be treated by something like WKB methods, but before spending a ton of time working that through I wanted to see if this kind of problem is well-known to the stack exchange community. Suggestions for how to proceed are very welcome.

• Thanks for the reference, I will take a look. As for your comment about Jacobi matrices - indeed, I am familiar with the recursion method, orthogonal polynomials, etc., and this is useful when looking at particular numerical problems. I'd like to move in a more analytic direction making use of the fact that the tight binding parameters behave smoothly at large $j$, where I think there should be a continuum limit. Commented Jan 3, 2023 at 14:45