Regular two-point boundary value problems for the Schr¨odingeroperator on a path E. Bendito, A. Carmona, A.M. Encinas and J.M. Gesto 1,2 Matem` atica Aplicada III Universitat Polit` ecnica de Catalunya Barcelona, Spain Abstract In this work we study the different type of regular boundary value problems on a path associated with the Schr¨ odinger 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 Cheby- shev polynomials since they verify a recurrence law similar to the one verified by the Sch¨ odinger operator on a path. Keywords: Discrete Schr¨odinger operator, Path, Boundary value problems, Green function, Chebyshev polynomials. 1 Introduction In this work, we analyze the linear boundary value problem in the context of the second order difference equation with constant coefficients associated with the Schr¨odinger operator on a finite path. Our study runs in parallel to 1 This work has been partly supported by the Spanish Research Council (Comisi´ on Inter- ministerial de Ciencia y Tecnolog´ ıa), under project BFM2003-06014. 2 Email: angeles.carmona@upc.edu Electronic Notes in Discrete Mathematics 28 (2007) 199–206 1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. www.elsevier.com/locate/endm doi:10.1016/j.endm.2007.01.027