Regular two-point boundary value problems for the Schrِdinger operator on a path

In this work we study the different type of regular boundary value problems on a path associated with the Schrِdinger operator. In particular, we obtain the Green function for each problem and we emphasize the case of Sturm-Liouville boundary conditions. In any case, the Green function is given in terms of second kind Chebyshev polynomials since they verify a recurrence law similar to the one verified by the Schِdinger operator on a path.