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

TL;DR

This paper introduces an exact solution to the discrete Schrödinger equation on graphs, providing a new effective model for quantum networks with applications in polymers and molecular chains.

Contribution

It presents a novel exact solution method for quantum graphs, enabling explicit eigenfunction and eigenvalue calculations for branched quantum networks.

Findings

Derived secular equation for arbitrary quantum graphs

Explicit eigenfunctions and eigenvalues for star graphs

Potential applications in conducting polymers and molecular chains

Abstract

We propose an approach to quantize discrete networks (graphs with discrete edges). We introduce a new exact solution of discrete Schrodinger 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 explicitely. Practical application of the model in conducting polymers and branched molecular chains is discussed.