Edge switching transformations of quantum graphs—a scattering approach

Schanz, H.; Smilansky, U.

doi:10.1090/spmj/1560

Edge switching transformations of quantum graphs—a scattering approach
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by H. Schanz and U. Smilansky

St. Petersburg Math. J. 30 (2019), 591-600

DOI: https://doi.org/10.1090/spmj/1560

Published electronically: April 12, 2019

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Abstract:

Some elementary transformations of quantum graphs are discussed and their effects on the spectra of the Schrödinger operators are studied. In particular, the edge swapping operation is considered, where the lengths of two edges are interchanged, as well as switching, where the connectivity of the graph is modified by reconnecting two edges. Both transformations preserve the total length of the graphs. This problem was already studied at length and in generality in a previous paper. Here, it is addressed from a different viewpoint based on a scattering approach, yielding a trace formula for the difference between the spectral counting functions before and after the transformation.

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Bibliographic Information

H. Schanz
Affiliation: Institute of Mechanical Engineering, University of Applied Sciences Magdeburg–Stendal, D-39114 Magdeburg, Germany
Email: holger.schanz@hs-magdeburg.de
U. Smilansky
Affiliation: Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
Email: uzy.smilansky@weizmann.ac.il
Received by editor(s): January 30, 2018
Published electronically: April 12, 2019

Dedicated:

Journal: St. Petersburg Math. J. 30 (2019), 591-600
DOI: https://doi.org/10.1090/spmj/1560
MathSciNet review: 3812008