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Nanosystems Physics Chemistry Mathematics

The Green function for simplest quantum graphs

K.K. Sabirov, U. A. Aminov, Kh.Sh. Saparov, M.K. Karimov, Kh. Abdikarimov

Published Dec 21, 2015• 1 citation
Publisher

Abstract

We treat the problem of the Green function for quantum graphs by focusing on such topologies as star and tree graphs.The exact Green function for the Schrödinger equation on primary star graphs is derived in the form of 3 × 3-matrix using the vertex boundary conditions providing continuity and current conservation.Extension of the approach for the derivation for the Green function on tree graph is presented.Possible practical applications of the obtained results are discussed.

DOI
10.17586/2220-8054-2015-6-6-762-766

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We treat the problem of the Green function for quantum graphs by focusing on such topologies as star and tree graphs.The exact Green function for the Schrödinger equation on primary star graphs is derived in the form of 3 × 3-matrix using the vertex boundary conditions providing continuity and current conservation.Extension of the approach for the derivation for the Green function on tree graph is presented.Possible practical applications of the obtained results are discussed.
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.
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.
We discuss some basic aspects of quantum fields on star graphs, focusing on boundary conditions, symmetries and scale invariance in particular. We investigate the four-fermion bulk interaction in detail. Using bosonization and vertex operators, we solve the model exactly for scale invariant boundary conditions, formulated in terms of the fermion current and without dissipation. The critical points are classified and determined explicitly. These results are applied for deriving the charge and spin transport, which have interesting physical features.
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