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arXiv:2411.14397 · Europhysics Letters (EPL)

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

M. Akramov, Carsten Trunk, Jambul Yusupov, Davron U. Matrasulov

Published Aug 29, 2024• 34 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1209/0295-5075/ad752e

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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.
We study the one-dimensional time-independent discrete Dirac equation on graphs with discrete edges. Both continuous and discrete formulations of the graphs are considered, with particular focus on star graphs with three edges as a simple example. Exact solutions for eigenstates and spectra are derived and compared with the continuum case, showing good agreement in tabulated results. This approach offers a systematic framework for analyzing relativistic quantum transport on branched structures relevant to nanostructures and quantum device design.
Abstract We consider PT-symmetric quantum graphs, in which the branching points provide PT-symmetric boundary conditions for the Schrödinger equation on a graph. For such branched quantum wires, we derive general boundary conditions, which keep the Hamiltonian as PT-symmetric with real eigenvalues and positively defined norm of the eigenfunctions. Secular equations for finding the eigenvalues of the quantum graph are derived. Breaking the Kirchhoff rule at the branching points in such systems is shown. Experimental realization of PT-symmetric quantum graphs on branched optical waveguides is discussed.
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Abstract One usually assumes that the Schrödinger operator on a thin quantum network (“fattened graph”) can be simulated by the ordinary Schrödinger operator on the corresponding one-dimensional quantum graph. On the other hand, each quantum graph can be constructed of standard elements – star graphs. We prove that a thin star-shaped quantum network which consists of a compact domain and a few straight semi-infinite quantum wires, with a two-dimensional Schrödinger operator on it, can be simulated by the corresponding solvable model in form of a one-dimensional star graph: an outer space, with an ordinary Schrödinger operator on the leads, a resonance vertex supplied with an inner space and a finite matrix in it and an appropriate boundary condition connecting the inner and outer components of elements from the domain of the model. The model is (locally) quantitatively consistent: the scattering matrix of the model on a certain spectral interval serves an approximation of the scattering matrix of the network. The role of the constructed star-graph model as a “jump-start” in analytic perturbation procedure on continuous spectrum is discussed.