Edge switching transformations of quantum graphs

M. Aizenman; H. Schanz; U. Smilansky; S. Warzel

doi:10.12693/APhysPolA.132.1699

Edge switching transformations of quantum graphs

M. Aizenman
, H. Schanz
, U. Smilansky
, S. Warzel

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

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 {E_n} _n ^∞ =1 and { Ẽ_n}_n ^∞ =1 correspondingly, are level-2 interlaced, so that E_n-2 ≤ Ẽ _n ≤ E_n+2. The proofs are guided by considerations of the quantum graphs' discrete analogs.

Original language	English (US)
Pages (from-to)	1699-1703
Number of pages	5
Journal	Acta Physica Polonica A
Volume	132
Issue number	6
DOIs	https://doi.org/10.12693/APhysPolA.132.1699
State	Published - Dec 2017

All Science Journal Classification (ASJC) codes

General Physics and Astronomy

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10.12693/APhysPolA.132.1699

Cite this

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title = "Edge switching transformations of quantum graphs",

abstract = "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 Schr{\"o}dinger 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 ≤ Ẽ n ≤ En+2. The proofs are guided by considerations of the quantum graphs' discrete analogs.",

author = "M. Aizenman and H. Schanz and U. Smilansky and S. Warzel",

note = "Funding Information: We thank Chris Joyner, and Rami Band for useful suggestions and corrections, and Mor Rozner for earlier numerical simulations which have guided us towards tighter results. The work of MA was supported in part by the NSF grant DMS-1613296 and the Weston Visiting Professorship at the Weizmann Institute. He thanks the Faculty of Mathematics and Computer Sciences and the Faculty of Physics at WIS for the hospitality enjoyed there.",

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doi = "10.12693/APhysPolA.132.1699",

language = "English (US)",

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T1 - Edge switching transformations of quantum graphs

AU - Aizenman, M.

AU - Schanz, H.

AU - Smilansky, U.

AU - Warzel, S.

N1 - Funding Information: We thank Chris Joyner, and Rami Band for useful suggestions and corrections, and Mor Rozner for earlier numerical simulations which have guided us towards tighter results. The work of MA was supported in part by the NSF grant DMS-1613296 and the Weston Visiting Professorship at the Weizmann Institute. He thanks the Faculty of Mathematics and Computer Sciences and the Faculty of Physics at WIS for the hospitality enjoyed there.

PY - 2017/12

Y1 - 2017/12

N2 - 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 ≤ Ẽ n ≤ En+2. The proofs are guided by considerations of the quantum graphs' discrete analogs.

AB - 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 ≤ Ẽ n ≤ En+2. The proofs are guided by considerations of the quantum graphs' discrete analogs.

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Edge switching transformations of quantum graphs

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