We aim here at analyzing self-adjoint boundary value problems on finite networks associated with positive semi-definite Schrödinger operators. In addition, we study the existence and uniqueness of solutions and its variational formulation. Moreover, we will tackle a well-known problem in the framework of Potential Theory, the so-called condenser principle. Then, we generalize of the concept of effective resistance between two vertices of the network and we characterize the Green function of some BVP in terms of effective resistances.