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Higher order differential operators on graphs
Stockholm University, Faculty of Science, Department of Mathematics.
2020 (English)Licentiate thesis, comprehensive summary (Other academic)
##### Abstract [en]

This thesis consists of two papers, enumerated by Roman numerals. The main focus is on the spectral theory of $n$-Laplacians. Here, an $n$-Laplacian, for integer $n$, refers to a metric graph equipped with a differential operator whose differential expression is the $2n$-th derivative.

In Paper I, a classification of all vertex conditions corresponding to self-adjoint $n$-Laplacians is given, and for these operators, a secular equation is derived. Their spectral asymptotics are analysed using the fact that the secular function is close to a trigonometric polynomial, a type of almost periodic function. The notion of the quasispectrum for $n$-Laplacians is introduced, identified with the positive roots of the associated trigonometric polynomial, and is proved to be unique. New results about almost periodic functions are proved, and using these it is shown that the quasispectrum asymptotically approximates the spectrum, counting multiplicities, and results about asymptotic isospectrality are deduced. The results obtained on almost periodic functions have wider applications outside the theory of differential operators.

Paper II deals more specifically with bi-Laplacians ($n=2$), and a notion of standard conditions is introduced. Upper and lower estimates for the spectral gap --- the difference between the two lowest eigenvalues - for these standard conditions are derived. This is achieved by adapting the methods of graph surgery used for quantum graphs to fourth order differential operators. It is observed that these methods offer stronger estimates for certain classes of metric graphs. A geometric version of the Ambartsumian theorem for these operators is proved.

##### Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2020. , p. 35
##### Keywords [en]
Almost periodic functions, differential operators on metric graphs, quantum graphs, estimation of eigenvalues
##### National Category
Mathematical Analysis
##### Identifiers
ISBN: 978-91-7797-988-3 (print)OAI: oai:DiVA.org:su-178070DiVA, id: diva2:1386786
##### Presentation
2020-02-03, 11:00 (English)
##### Funder
Swedish Research Council, D0497301Available from: 2020-01-20 Created: 2020-01-20 Last updated: 2020-01-31Bibliographically approved
##### List of papers
1. Higher order differential operators on graphs and almost periodic functions
Open this publication in new window or tab >>Higher order differential operators on graphs and almost periodic functions
2020 (English)Report (Other academic)
##### Abstract [en]

The spectra of $n$-Laplacian operators $(-\Delta)^n$ on finite metric graphs are studied. An effective secular equation is derived and the spectral asymptotics analysed, exploiting the fact that the secular function is close to a trigonometric polynomial. The notion of the quasispectrum is introduced and its uniqueness is proved using the theory of almost periodic functions. To achieve this, new results concerning roots of functions close to almost periodic functions are proved. The results obtained on almost periodic functions are of general interest outside the theory of differential operators.

##### Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2020. p. 67
##### Series
Research Reports in Mathematics, ISSN 1401-5617 ; 1
##### Keywords
Almost periodic functions, quantum graphs
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:su:diva-178014 (URN)
##### Funder
Swedish Research Council, D0497301 Available from: 2020-01-16 Created: 2020-01-16 Last updated: 2020-02-06Bibliographically approved
2. On the spectral gap for networks of beams
Open this publication in new window or tab >>On the spectral gap for networks of beams
2020 (English)Report (Other academic)
##### Abstract [en]

A notion of standard vertex conditions for beam operators (the fourth derivative) on metric graphs is presented, and the spectral gap (the difference between the first two eigenvalues) for the operator with these conditions is studied. Upper and lower estimates for the spectral gap are obtained, and it is shown that stronger estimates can be obtained for certain classes of graphs. Graph surgery is used as a technique for estimation. A geometric version of the Ambartsumian theorem for networks of beams is proved.

##### Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2020. p. 17
##### Series
Research Reports in Mathematics, ISSN 1401-5617 ; 2
##### Keywords
Bi-Laplacians, quantum graphs, graph surgery, spectral geometry of quantum graphs, bounds on spectral gaps
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:su:diva-178015 (URN)
##### Funder
Swedish Research Council, D0497301 Available from: 2020-01-16 Created: 2020-01-16 Last updated: 2020-02-06Bibliographically approved

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Cite
Citation style
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