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Dissertation/Thesis Abstract

Quantum graphs and their spectra
by Rueckriemen, Ralf, Ph.D., Dartmouth College, 2011, 85; 3494278
Abstract (Summary)

We show that families of leafless quantum graphs that are isospectral for the standard Laplacian are finite. We show that the minimum edge length is a spectral invariant. We give an upper bound for the size of isospectral families in terms of the total edge length of the quantum graphs.

We define the Bloch spectrum of a quantum graph to be the map that assigns to each element in the deRham cohomology the spectrum of an associated magnetic Schrödinger operator. We show that the Bloch spectrum determines the Albanese torus, the block structure and the planarity of the graph. It determines a geometric dual of a planar graph. This enables us to show that the Bloch spectrum identifies and completely determines planar 3-connected quantum graphs.

Indexing (document details)
Advisor: Gordon, Carolyn S.
Commitee: Kuchment, Peter, Sutton, Craig, Winkler, Pete
School: Dartmouth College
Department: Mathematics
School Location: United States -- New Hampshire
Source: DAI-B 73/05, Dissertation Abstracts International
Subjects: Applied Mathematics, Mathematics
Keywords: Inverse spectra, Laplace operator on quantum graphs, Quantum graphs, Spectral geometry
Publication Number: 3494278
ISBN: 978-1-267-15715-7
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