# References on discrete Sturm-Liouville eigenvectors convergence

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 ?