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

¹ National University of Uzbekistan - Universitet Str. 4, 100174, Tashkent, Uzbekistan
² Institute for Mathematics, TU Ilmenau - Weimarer Str. 25, 98693, Ilmenau, Germany
³ Kimyo International University in Tashkent - Usman Nasyr Str. 156, 100121, Tashkent, Uzbekistan
⁴ Turin Polytechnic University in Tashkent - Niyazov Str. 17, 100095, Tashkent, Uzbekistan
⁵ Center for Theoretical Physics, Khazar University - 41 Mehseti Street, Baku, AZ1096, Azerbaijan

Received: 18 April 2024
Accepted: 29 August 2024

Abstract

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