Discrete Schrödinger equation on graphs: An effective model for branched quantum lattice

Abstract

We propose an approach to quantize discrete networks (graphs with discrete edges). We introduce a new exact solution of discrete Schrödinger equation that is used to write the solution for quantum graphs. Formulation of the problem and derivation of secular equation for arbitrary quantum graphs is presented. Application of the approach for the star graph is demonstrated by obtaining eigenfunctions and eigenvalues explicitly. Practical application of the model in conducting polymers and branched molecular chains is discussed.