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Spectral estimates and Ambartsumian-type theorems for quantum graphs
Stockholm University, Faculty of Science, Department of Mathematics. (Analysis)
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers and deals with the spectral theory of quantum graphs. A quantum graph is a metric graph equipped with a self-adjoint Schrödinger operator acting on functions defined on the edges of the graph subject to certain vertex conditions.

In Paper I we establish a spectral estimate implying that the distance between the eigenvalues of a Laplace and a Schrödinger operator on the same graph is bounded by a constant depending only on the graph and the integral of the potential. We use this to generalize a geometric version of Ambartsumian's Theorem to the case of Schrödinger operators with standard vertex conditions.

In Paper II we extend the results of Paper I to more general vertex conditions but also provide explicit examples of quantum graphs that show that the results are not valid for all allowed vertex conditions.

In Paper III the zero sets of almost periodic functions are investigated, and it is shown that if two functions have zeros that are asymptotically close, they must coincide. This is relevant to the spectral theory of quantum graphs as the eigenvalues of a quantum graph are given by the zeros of a trigonometric polynomial, which is almost periodic.

In Paper IV we give a proof of the result in Paper III which does not rely on the theory of almost periodic functions and apply this to show that asymptotically isospectral quantum graphs are in fact isospectral. This allows us to generalize two uniqueness results in the spectral theory of quantum graphs: we show that if the spectrum of a Schrödinger operator with standard vertex conditions on a graph is equal to the spectrum of a Laplace operator on another graph then the potential must be zero, and we show that a metric graph with rationally independent edge-lengths is uniquely determined by the spectrum of a Schrödinger operator with standard vertex conditions on the graph.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2018.
Keywords [en]
Spectral estimates, quantum graphs, Ambartsumian, trigonometric polynomials
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-155098ISBN: 978-91-7797-292-1 (print)ISBN: 978-91-7797-293-8 (print)OAI: oai:DiVA.org:su-155098DiVA, id: diva2:1196937
Public defence
2018-05-31, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Manuscript. Paper 3: Manuscript. Paper 4: Manuscript.

Available from: 2018-05-07 Created: 2018-04-11 Last updated: 2018-04-27Bibliographically approved
List of papers
1. Schrödinger Operators on Graphs and Geometry II. Integrable Potentials and an Ambartsumian Theorem
Open this publication in new window or tab >>Schrödinger Operators on Graphs and Geometry II. Integrable Potentials and an Ambartsumian Theorem
2016 (English)Report (Other academic)
Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2016. p. 29
Series
Research Reports in Mathematics, ISSN 1401-5617 ; 1
Keywords
Ambartsumian Theorem, Euler Characteristic, inverse problem, metric graph, quantum graph, Schrödinger operator
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-134729 (URN)
Available from: 2016-10-17 Created: 2016-10-17 Last updated: 2018-04-11Bibliographically approved
2. Schrödinger operators on graphs: a geometric version of the Ambartsumian theorem
Open this publication in new window or tab >>Schrödinger operators on graphs: a geometric version of the Ambartsumian theorem
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Schrödinger operators on metric graphs with general vertex conditions are studied. Explicit spectral asymptotics is derived in terms of the spectrum of standard Laplacians.

A geometric version of Ambartsumian theorem is proven under the assumption that the vertex conditions are asymptotically properly connecting and asymptotically standard. By constructing explicit counterexamples it is shown that the geometric Albartsumian theorem does not hold in general without additional assumptions on the vertex conditions.

Keywords
Quantum graphs, Ambartsumian theorem, vertex conditions
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-154992 (URN)
Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-12Bibliographically approved
3. A note on asymptotically close zeros of almost periodic functions
Open this publication in new window or tab >>A note on asymptotically close zeros of almost periodic functions
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The zero sets of almost periodic functions --- i.e. uniform limits of trigonometric polynomials --- are investigated. It is shown that if two almost periodic functions $f_1$ and $f_2$ have zeros that are asymptotically close then all their zeros have to coincide.

Keywords
Almost periodic functions, almost periodic sets, quantum graphs.
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-154993 (URN)
Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-12Bibliographically approved
4. Asymptotically isospectral quantum graphs and trigonometric polynomials
Open this publication in new window or tab >>Asymptotically isospectral quantum graphs and trigonometric polynomials
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The theory of almost periodic functions is used to investigate spectral properties of Schr\"odinger operators on metric graphs, also known as quantum graphs. In particular we prove that two Schr\"odinger operators may have asymptotically close spectra if and only if the corresponding Laplacians are isospectral. The case of general vertex conditions and integrable potentials is considered. In particular, our result implies that a Schr\"odinger operator is isospectral to the standard Laplacian on a may be different metric graph only if the potential is identically equal to zero.

Keywords
Quantum graphs, almost periodic functions
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-154994 (URN)
Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-12Bibliographically approved

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