Let $ L : u_n \mapsto a_n u_{n + 1} + b_n u_n + a_{n - 1} u_{n -1} = \nabla ( a_n \Delta u_n ) + (b_n + a_n + a_{n - 1}) u_n $ be a discrete Sturm-Liouville operator, with $ \nabla u_n := u_{n + 1} - u_n $ and $ \Delta u_n := u_n - u_{n -1} $. Let us suppose that $ L $ depends on a parameter $ t $ that tends to $ +\infty $ and that with a good scaling $ n = [\alpha(t) x + \beta(t)] $, one has convergence of $ L $ towards a continuous Sturm-Liouville operator $ \mathbb{L} : f \mapsto (Af')' + B $ where $ A $ and $ B $ are good functions (for instance polynomials, if it can help). I precise that I have Dirichlet boundary conditions $ u_0 = 0 $ and that one can suppose $ L $ to be self-adjoint for a good scalar product, for instance $ \ell^2(\mathbb{N}) $.

I am looking for results of convergence of eigenvectors of $ L $ towards eigenvectors of $ \mathbb{L} $. Are there any references, a theory, etc. ? Some simple examples with explicit computations are welcome, for instance convergence towards some hypergeometric operator $ \mathbb{L} $. In fact, are there references on these discrete Sturm-Liouville operators ?