We aim here at analyzing the fundamental properties of positive semidefinite Schrödinger operators on networks. We show that such operators correspond to perturbations of the combinatorial Laplacian through 0-order terms that can be totally negative on a proper subset of the network. In addition, we prove that these discrete operators have analogous properties to the ones of elliptic second order operators on Riemannian manifolds, namely the monotonicity, the minimum principle, the variational treatment of Dirichlet problems and the condenser principle. Unlike the continuous case, a discrete Schrödinger operator can be interpreted as an integral operator and therefore a discrete Potential Theory with respect to its associated kernel can be built. We prove that the Schrödinger kernel satisfies enough principles to assure the existence of equilibrium measures for any proper subset. These measures are used to obtain systematic expressions of the Green and Poisson kernels associated with Dirichlet problems.

The purpose of this paper is to construct solutions of self-adjoint boundary value problems on finite networks. To this end, we obtain explicit expressions of the Green functions for all different boundary value problems. The method consists of reducing each boundary value problem either to a Dirichlet problem or to a Poisson equation on a new network closely related with the former boundary value problem. In this process we also get an explicit expression of the Poisson kernel for the Dirichlet problem. In all cases, we express the Green function in terms of equilibrium measures solely, which can be obtained as the unique solution of linear programming problems. In particular, we get analytic expressions of the Green function for the following problems: the Poisson equation on a distance-regular graph, the Dirichlet problem on an infinite distance-regular graph, and the Neumann problem on a ball of an homogeneous tree. Academic Press

ABSTRACT:We show here that the Kirchhoff index of a network is the average of the Wiener capacities of its vertices. Moreover, we obtain a closed-form formula for the effective resistance between any pair of vertices when the considered network has some symmetries, which allows us to give the corresponding formulas for the Kirchhoff index. In addition, we find the expression for the Foster's n-th formula.

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In this work we analyze the boundary value problems on a path associated with Schrödinger operators with constant ground state. These problems include the cases in which the boundary has two, one or none vertices. In addition, we study the periodic boundary value problem that corresponds to the Poisson equation in a cycle. Moreover, we obtain the Green's function for each regular problem and the eigenvalues and their corresponding eigenfunctions otherwise. In each case, the Green's functions, the eigenvalues and the eigenfunctions are given in terms of first, second and third kind Chebyshev polynomials.