Dynamics of Nodal Points and the Nodal Count on a Family of Quantum Graphs

Ram Band; Gregory Berkolaiko; Uzy Smilansky

doi:10.1007/s00023-011-0124-1

Dynamics of Nodal Points and the Nodal Count on a Family of Quantum Graphs

Ram Band, Gregory Berkolaiko, Uzy Smilansky

Department of Physics of Complex Systems

Weizmann Institute of Science

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

Abstract

We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schrödinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive several formulas relating the number of the zeros of the n-th eigenfunction to the spectrum of the graph and of some of its subgraphs. In a special case of the so-called dihedral graph we prove an explicit formula that only uses the lengths of the edges, entirely bypassing the information about the graph's eigenvalues. The results are explained from the point of view of the dynamics of zeros of the solutions to the scattering problem.

Original language	English
Pages (from-to)	145-184
Number of pages	40
Journal	Annales Henri Poincare
Volume	13
Issue number	1
DOIs	https://doi.org/10.1007/s00023-011-0124-1
State	Published - Feb 2012

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Nuclear and High Energy Physics
Mathematical Physics

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10.1007/s00023-011-0124-1

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title = "Dynamics of Nodal Points and the Nodal Count on a Family of Quantum Graphs",

abstract = "We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schr{\"o}dinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive several formulas relating the number of the zeros of the n-th eigenfunction to the spectrum of the graph and of some of its subgraphs. In a special case of the so-called dihedral graph we prove an explicit formula that only uses the lengths of the edges, entirely bypassing the information about the graph's eigenvalues. The results are explained from the point of view of the dynamics of zeros of the solutions to the scattering problem.",

author = "Ram Band and Gregory Berkolaiko and Uzy Smilansky",

note = "Funding Information: It is a pleasure to acknowledge Sven Gnutzmann for fruitful discussions about the scattering matrix properties. We are grateful to Peter Kuchment for suggesting to extend Theorem 4.1 to what is now Theorem 4.2. We also wish to thank Amit Godel for the careful examination of the proof of Theorem 3.3. The work was supported by the Minerva Center for Nonlinear Physics, the Einstein (Minerva) Center at the Weizmann Institute and the Wales Institute of Mathematical and Computational Sciences) (WIMCS). Grants from Funding Information: EPSRC (grant EP/G021287), ISF (grant 166/09), BSF (grant 2006065) and NSF (DMS-0604859 and DMS-0907968) are acknowledged.",

year = "2012",

month = feb,

doi = "10.1007/s00023-011-0124-1",

language = "الإنجليزيّة",

volume = "13",

pages = "145--184",

journal = "Annales Henri Poincare",

issn = "1424-0637",

publisher = "Birkhauser Verlag Basel",

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AU - Band, Ram

AU - Berkolaiko, Gregory

AU - Smilansky, Uzy

N1 - Funding Information: It is a pleasure to acknowledge Sven Gnutzmann for fruitful discussions about the scattering matrix properties. We are grateful to Peter Kuchment for suggesting to extend Theorem 4.1 to what is now Theorem 4.2. We also wish to thank Amit Godel for the careful examination of the proof of Theorem 3.3. The work was supported by the Minerva Center for Nonlinear Physics, the Einstein (Minerva) Center at the Weizmann Institute and the Wales Institute of Mathematical and Computational Sciences) (WIMCS). Grants from Funding Information: EPSRC (grant EP/G021287), ISF (grant 166/09), BSF (grant 2006065) and NSF (DMS-0604859 and DMS-0907968) are acknowledged.

PY - 2012/2

Y1 - 2012/2

N2 - We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schrödinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive several formulas relating the number of the zeros of the n-th eigenfunction to the spectrum of the graph and of some of its subgraphs. In a special case of the so-called dihedral graph we prove an explicit formula that only uses the lengths of the edges, entirely bypassing the information about the graph's eigenvalues. The results are explained from the point of view of the dynamics of zeros of the solutions to the scattering problem.

AB - We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schrödinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive several formulas relating the number of the zeros of the n-th eigenfunction to the spectrum of the graph and of some of its subgraphs. In a special case of the so-called dihedral graph we prove an explicit formula that only uses the lengths of the edges, entirely bypassing the information about the graph's eigenvalues. The results are explained from the point of view of the dynamics of zeros of the solutions to the scattering problem.

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U2 - 10.1007/s00023-011-0124-1

DO - 10.1007/s00023-011-0124-1

M3 - مقالة

SN - 1424-0637

VL - 13

SP - 145

EP - 184

JO - Annales Henri Poincare

JF - Annales Henri Poincare

IS - 1

ER -

Dynamics of Nodal Points and the Nodal Count on a Family of Quantum Graphs

Abstract

All Science Journal Classification (ASJC) codes

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