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Journal of Physics A Mathematical and General

Green function approach for general quantum graphs

Alexandre G. M. Schmidt, Bin Kang Cheng, M. G. E. da Luz

Published Oct 8, 2003• 28 citations
Publisher

Abstract

We present a recursive prescription to calculate the exact Green function for general quantum graphs. For the closed case, the expression for the poles of G —which gives the individual eigenstates—has the structure of a semiclassical formula, where the sum over the periodic orbit is already performed. As applications we discuss eigenstate localization for a three-arm closed star and filter-like mechanisms for transmission throughout an open trident graph.

DOI
10.1088/0305-4470/36/42/l01

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We present a recursive prescription to calculate the exact Green function for general quantum graphs. For the closed case, the expression for the poles of G —which gives the individual eigenstates—has the structure of a semiclassical formula, where the sum over the periodic orbit is already performed. As applications we discuss eigenstate localization for a three-arm closed star and filter-like mechanisms for transmission throughout an open trident graph.
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Recently, general point interactions in one dimension has been used to model a large number of different phenomena in quantum mechanics. Such potentials, however, require some sort of regularization to lead to meaningful results. The usual ways to do so rely on technicalities that may hide important physical aspects of the problem. In this work we present a method to calculate the exact Green functions for general point interactions in one dimension. Our approach differs from previous ones because it is based only on physical quantities, namely, the scattering coefficients R and T to construct G. Renormalization or particular mathematical prescriptions are not invoked. The simple formulation of the method makes it easy to extend to more general contexts, such as for lattices of N general point interactions, on a line, on a half-line, under periodic boundary conditions, and confined in a box.