Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Schrödinger Operators on Graphs and Geometry II. Integrable Potentials and an Ambartsumian Theorem
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
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
Keyword [en]
Ambartsumian Theorem, Euler Characteristic, inverse problem, metric graph, quantum graph, Schrödinger operator
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-134729OAI: oai:DiVA.org:su-134729DiVA, id: diva2:1037650
Available from: 2016-10-17 Created: 2016-10-17 Last updated: 2018-04-11Bibliographically approved
In thesis
1. Spectral estimates and Ambartsumian-type theorems for quantum graphs
Open this publication in new window or tab >>Spectral estimates and Ambartsumian-type theorems for quantum graphs
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
Keyword
Spectral estimates, quantum graphs, Ambartsumian, trigonometric polynomials
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-155098 (URN)978-91-7797-292-1 (ISBN)978-91-7797-293-8 (ISBN)
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

Open Access in DiVA

fulltext(381 kB)78 downloads
File information
File name FULLTEXT01.pdfFile size 381 kBChecksum SHA-512
ea93571904bd30fed863f9cb248578fa557d093af5a0076d8786debd462fb0b51223fa1e522dea95d25baecc05d4a71afa9f9f93346276a548ea42f7b0eaf553
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Boman, JanKurasov, PavelSuhr, Rune
By organisation
Department of Mathematics
Mathematical Analysis

Search outside of DiVA

GoogleGoogle Scholar
Total: 78 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

urn-nbn

Altmetric score

urn-nbn
Total: 62 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf