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Edited by: Luiz A. Manzoni, Concordia College, United States

Reviewed by: Fabio Rinaldi, Universitá degli Studi Guglielmo Marconi, Italy; Fabiano Andrade, Universidade Estadual de Ponta Grossa, Brazil

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The exact Green function is constructed for a quantum system, with known Green function, which is decorated by two delta function impurities. It is shown that when two such impurities coincide they behave as a single singular potential with combined amplitude. The results are extended to N impurities and higher dimensions.

The one dimensional harmonic oscillator or square well, for example, for which the energy -dependent Green function

In this note a corresponding formula is derived for the case

We first note that the same argument can be used for the time dependent-, as well as the energy-dependent Green functions, so we shall omit the third argument and write simply

Beginning with the Dyson equation, noting that _{0}(_{0}(

where the integration extends over the system domain, one has the set of equations

The linear Equations (4) and (5) are easily solved for

with

By inserting (6) and (7) into (3) we obtain the desired expression

By setting μ to 0 (9) reduces to (1), proving this expression as well. The most salient feature of (9) is the denominator

Two further points can be made. Nothing in the derivation of (9) restricts it to the line. If we accept the standard definition

A second observation is that

there will be

_{0}(_{j}δ(_{j}),

_{lm} = δ_{lm}−λ_{l}_{0}(_{l}, _{m}).

Thus,

which reduces to the

Note that if all the λs and

Equation (11) might offer a new approach to Kronig-Penney-type systems for periodic or random unit cells.

Finally, it should be pointed out that the work in this note is paralleled in the theory of quantum graphs introduced by Linus Pauling about 1930 to describe electrons in molecules which has developed into a sophisticated and important branch of quantum physics [

The author confirms being the sole contributor of this work and has approved it for publication.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author thanks Prof. S. Fassari and Prof. L. M. Nieto for helpful comments and acknowledges the financial support of MINECO (Project MTM2014-57129-C2-1-P) and Junta de Castilla y Leon (VA057U16). The author thanks a referee for pointing out the relevance to quantum graphs.