Edge Switching Transformations of Quantum Graphs

M. Aizenman a, H. Schanz b, U. Smilansky c, S. Warzel d aDepartments of Physics and Mathematics, Princeton University, Princeton NJ 08540, USA bInstitute of Mechanical Engineering, University of Applied Sciences Magdeburg-Stendal, D-39114 Magdeburg, Germany cDepartment of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel dZentrum Mathematik, TU München, Boltzmannstr. 3, 85747 Garching, Germany

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Discussed here are the effects of basics graph transformations on the spectra of associated quantum graphs. In particular it is shown that under an edge switch the spectrum of the transformed Schrödinger operator is interlaced with that of the original one. By implication, under edge swap the spectra before and after the transformation, denoted by {En}n=1∞ and {Ẽn}n=1∞ correspondingly, are level-2 interlaced, so that En-2 ≤ Ẽn ≤ En+2. The proofs are guided by considerations of the quantum graphs' discrete analogs.

DOI: 10.12693/APhysPolA.132.1699 PACS numbers: 05.45.Mt, 02.10.Ox