Change search
Link to record
Permanent link

Direct link
Publications (10 of 68) Show all publications
Berkolaiko, G., Kennedy, J. B., Kurasov, P. & Mugnolo, D. (2023). Impediments to diffusion in quantum graphs: Geometry-based upper bounds on the spectral gap. Proceedings of the American Mathematical Society, 151, 3439-3455
Open this publication in new window or tab >>Impediments to diffusion in quantum graphs: Geometry-based upper bounds on the spectral gap
2023 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 151, p. 3439-3455Article in journal (Refereed) Published
Abstract [en]

We derive several upper bounds on the spectral gap of the Laplacian on compact metric graphs with standard or Dirichlet vertex conditions. In particular, we obtain estimates based on the length of a shortest cycle (girth), diameter, total length of the graph, as well as further metric quantities introduced here for the first time, such as the avoidance diameter. Using known results about Ramanujan graphs, a class of expander graphs, we also prove that some of these metric quantities, or combinations thereof, do not to deliver any spectral bounds with the correct scaling.

Keywords
Quantum graphs, Girth, Spectral geometry of quantum graphs, Bounds on spectral gaps
National Category
Discrete Mathematics
Identifiers
urn:nbn:se:su:diva-217358 (URN)10.1090/proc/16322 (DOI)000982496000001 ()2-s2.0-85162213019 (Scopus ID)
Available from: 2023-05-29 Created: 2023-05-29 Last updated: 2023-10-10Bibliographically approved
Kurasov, P. (2023). On Crossroads of Spectral Theory with Sergey Naboko. In: Malcolm Brown; Fritz Gesztesy; Pavel Kurasov; Ari Laptev; Barry Simon; Gunter Stolz; Ian Wood (Ed.), From Complex Analysis to Operator Theory: A Panorama: In Memory of Sergey Naboko (pp. 43-48). Cham: Birkhäuser Verlag
Open this publication in new window or tab >>On Crossroads of Spectral Theory with Sergey Naboko
2023 (English)In: From Complex Analysis to Operator Theory: A Panorama: In Memory of Sergey Naboko / [ed] Malcolm Brown; Fritz Gesztesy; Pavel Kurasov; Ari Laptev; Barry Simon; Gunter Stolz; Ian Wood, Cham: Birkhäuser Verlag, 2023, p. 43-48Chapter in book (Refereed)
Abstract [en]

Several recent achievements of Sergey Naboko in spectral theory of singular differential operators and metric graphs are described. The impact of Sergey’s work on my own research career is underlined.

Place, publisher, year, edition, pages
Cham: Birkhäuser Verlag, 2023
Series
Operator Theory: Advances and Applications, ISSN 0255-0156, E-ISSN 2296-4878 ; 291
Keywords
Embedded eigenvalues, Hain-Lüst operator, Metric graphs, Wigner-von Neumann potentials
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:su:diva-223049 (URN)10.1007/978-3-031-31139-0_4 (DOI)2-s2.0-85172476187 (Scopus ID)978-3-031-31138-3 (ISBN)978-3-031-31139-0 (ISBN)
Available from: 2023-10-18 Created: 2023-10-18 Last updated: 2023-10-18Bibliographically approved
Ławniczak, M., Kurasov, P., Bauch, S., Białous, M., Akhshani, A. & Sirko, L. (2021). A new spectral invariant for quantum graphs. Scientific Reports, 11(1), Article ID 15342.
Open this publication in new window or tab >>A new spectral invariant for quantum graphs
Show others...
2021 (English)In: Scientific Reports, E-ISSN 2045-2322, Vol. 11, no 1, article id 15342Article in journal (Refereed) Published
Abstract [en]

The Euler characteristic i.e., the difference between the number of vertices |V| and edges |E| is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary to introduce a new spectral invariant, the generalized Euler characteristic chi G:=|V|-|VD|-|E|, with |VD| denoting the number of Dirichlet vertices. We demonstrate theoretically and experimentally that the generalized Euler characteristic chi G of quantum graphs and microwave networks can be determined from small sets of lowest eigenfrequencies. If the topology of the graph is known, the generalized Euler characteristic chi G can be used to determine the number of Dirichlet vertices. That makes the generalized Euler characteristic chi G a new powerful tool for studying of physical systems modeled by differential equations on metric graphs including isoscattering and neural networks where both Neumann and Dirichlet boundary conditions occur.

National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-197696 (URN)10.1038/s41598-021-94331-0 (DOI)000683353600003 ()34321508 (PubMedID)
Available from: 2021-10-14 Created: 2021-10-14 Last updated: 2022-09-15Bibliographically approved
Kennedy, J. B., Kurasov, P., Léna, C. & Mugnolo, D. (2021). A theory of spectral partitions of metric graphs. Calculus of Variations and Partial Differential Equations, 60(2), Article ID 61.
Open this publication in new window or tab >>A theory of spectral partitions of metric graphs
2021 (English)In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 60, no 2, article id 61Article in journal (Refereed) Published
Abstract [en]

We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in Band et al. (Commun Math Phys 311:815-838, 2012) as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic-rather than numerical-results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in Conti et al. (Calc Var 22:45-72, 2005), Helffer et al. (Ann Inst Henri Poincare Anal Non Lineaire 26:101-138, 2009), but we can also generalise some of them and answer (the graph counterparts of) a few open questions.

Keywords
34B45, 35P15, 49Q10, 81Q35
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-193046 (URN)10.1007/s00526-021-01966-y (DOI)000635939300001 ()
Available from: 2021-05-09 Created: 2021-05-09 Last updated: 2022-02-25Bibliographically approved
Kurasov, P. (2021). Always Detectable Eigenfunctions on Metric Graphs. Acta Physica Polonica. A, 140(6), 510-513
Open this publication in new window or tab >>Always Detectable Eigenfunctions on Metric Graphs
2021 (English)In: Acta Physica Polonica. A, ISSN 0587-4246, E-ISSN 1898-794X, Vol. 140, no 6, p. 510-513Article in journal (Refereed) Published
Abstract [en]

It is proven that Laplacians with standard vertex continuous on metric trees and with standard and Dirichlet conditions on arbitrary metric graphs possess an infinite sequence of simple eigenvalues with the eigenfunctions not equal to zero in any non-Dirichlet vertex.

National Category
Physical Sciences
Identifiers
urn:nbn:se:su:diva-201119 (URN)10.12693/APhysPolA.140.510 (DOI)000740780200005 ()
Available from: 2022-01-18 Created: 2022-01-18 Last updated: 2022-01-18Bibliographically approved
Kurasov, P., Mugnolo, D. & Wolf, V. (2021). Analytic solutions for stochastic hybrid models of gene regulatory networks. Journal of Mathematical Biology, 82(1-2), Article ID 9.
Open this publication in new window or tab >>Analytic solutions for stochastic hybrid models of gene regulatory networks
2021 (English)In: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 82, no 1-2, article id 9Article in journal (Refereed) Published
Abstract [en]

Discrete-state stochastic models are a popular approach to describe the inherent stochasticity of gene expression in single cells. The analysis of such models is hindered by the fact that the underlying discrete state space is extremely large. Therefore hybrid models, in which protein counts are replaced by average protein concentrations, have become a popular alternative. The evolution of the corresponding probability density functions is given by a coupled system of hyperbolic PDEs. This system has Markovian nature but its hyperbolic structure makes it difficult to apply standard functional analytical methods. We are able to prove convergence towards the stationary solution and determine such equilibrium explicitly by combining abstract methods from the theory of positive operators and elementary ideas from potential analysis.

Keywords
Petri networks, Systems of PDEs, Positive C-0-semigroups, Piecewise deterministic Markov processes
National Category
Biological Sciences
Identifiers
urn:nbn:se:su:diva-191338 (URN)10.1007/s00285-021-01549-7 (DOI)000613325500001 ()33496854 (PubMedID)
Available from: 2021-03-15 Created: 2021-03-15 Last updated: 2022-02-25Bibliographically approved
Ławniczak, M., Kurasov, P., Bauch, S., Białous, M. & Sirko, L. (2021). Euler Characteristic of Graphs and Networks. Paper presented at XLVI Extraordinary Congress of Polish Physicists, Warsaw, Poland, October 16–18, 2020. Acta Physica Polonica. A, 139(3), 323-327
Open this publication in new window or tab >>Euler Characteristic of Graphs and Networks
Show others...
2021 (English)In: Acta Physica Polonica. A, ISSN 0587-4246, E-ISSN 1898-794X, Vol. 139, no 3, p. 323-327Article in journal (Refereed) Published
Abstract [en]

The Euler characteristic chi = vertical bar V vertical bar - vertical bar E vertical bar is an important topological characteristic of graphs and networks. Here, vertical bar V vertical bar and vertical bar E vertical bar denote the number of vertices and edges of a graph or a network. It has been shown in [Phys. Rev. E 101, 052320 (2020)] that the Euler characteristic can be determined from a finite sequence of the lowest eigenenergies lambda(1), ..., lambda(N) of a simple quantum graph. We will test this finding numerically, using chaotic graphs with vertical bar V vertical bar = 8 vertices. We will consider complete (fully connected) and incomplete realizations of 8-vertex graphs. The properties of the Euler characteristic will also be tested experimentally using the sequence of the lowest resonances of the 5-vertex microwave network. We will show that the Euler characteristic chi can be used to reveal whether the graph is planar or not.

National Category
Physical Sciences
Identifiers
urn:nbn:se:su:diva-194547 (URN)10.12693/APhysPolA.139.323 (DOI)000637753700022 ()
Conference
XLVI Extraordinary Congress of Polish Physicists, Warsaw, Poland, October 16–18, 2020
Available from: 2021-08-02 Created: 2021-08-02 Last updated: 2022-02-25Bibliographically approved
Kurasov, P. & Muller, J. (2021). n-Laplacians on Metric Graphs and Almost Periodic Functions: I. Annales de l'Institute Henri Poincare. Physique theorique, 22(1), 121-169
Open this publication in new window or tab >>n-Laplacians on Metric Graphs and Almost Periodic Functions: I
2021 (English)In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 22, no 1, p. 121-169Article in journal (Refereed) Published
Abstract [en]

The spectra of n-Laplacian operators (−Δ)n on finite metric graphs are studied. An effective secular equation is derived and the spectral asymptotics are 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.

National Category
Physical Sciences Mathematics
Identifiers
urn:nbn:se:su:diva-189361 (URN)10.1007/s00023-020-00979-1 (DOI)000591565800001 ()
Available from: 2021-01-20 Created: 2021-01-20 Last updated: 2022-02-25Bibliographically approved
Bauch, S., Ławniczak, M., Wrochna, J., Kurasov, P. & Sirko, L. (2021). Some Applications of Generalized Euler Characteristic of Quantum Graphs and Microwave Networks. Paper presented at 10th Workshop on Quantum Chaos and Localisation Phenomena, Warsaw, Poland (online), May 27-28, 2021. Acta Physica Polonica. A, 140(6), 525-531
Open this publication in new window or tab >>Some Applications of Generalized Euler Characteristic of Quantum Graphs and Microwave Networks
Show others...
2021 (English)In: Acta Physica Polonica. A, ISSN 0587-4246, E-ISSN 1898-794X, Vol. 140, no 6, p. 525-531Article in journal (Refereed) Published
Abstract [en]

In this article we continue to explore the possibilities offered by our discovery that one of the main graph and network characteristic, the generalized Euler characteristic iG, can be determined from a graph/network spectrum. We show that using the generalized Euler characteristic iG the number of vertices with Dirichlet |VD| boundary conditions of a family of graphs/networks created on the basis of the standard quantum graphs or microwave networks can be easily identified. We also present a new application of the generalized Euler characteristic for checking the completeness of graphs/networks spectra in the low energy range.

Keywords
quantum graphs, Euler characteristic, boundary conditions, microwave networks
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-201285 (URN)10.12693/APhysPolA.140.525 (DOI)000740780200007 ()
Conference
10th Workshop on Quantum Chaos and Localisation Phenomena, Warsaw, Poland (online), May 27-28, 2021
Available from: 2022-01-24 Created: 2022-01-24 Last updated: 2022-01-24Bibliographically approved
Kurasov, P. & Suhr, R. (2020). Asymptotically isospectral quantum graphs and generalised trigonometric polynomials. Journal of Mathematical Analysis and Applications, 488(1), Article ID 124049.
Open this publication in new window or tab >>Asymptotically isospectral quantum graphs and generalised trigonometric polynomials
2020 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 488, no 1, article id 124049Article in journal (Refereed) Published
Abstract [en]

The theory of almost periodic functions is used to investigate spectral properties of Schrodinger operators on metric graphs, also known as quantum graphs. In particular we prove that two Schrodinger operators may have asymptotically close spectra if and only if the corresponding reference Laplacians are isospectral. Our result implies that a Schrodinger 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
Identifiers
urn:nbn:se:su:diva-181896 (URN)10.1016/j.jmaa.2020.124049 (DOI)000525911000001 ()
Available from: 2020-08-10 Created: 2020-08-10 Last updated: 2022-03-23Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-3256-6968

Search in DiVA

Show all publications