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  • 1. Albeverio, Sergio
    et al.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Singular perturbations of differential operators: solvable Schrödinger type operators2000Book (Refereed)
  • 2. Astudillo, Maria
    et al.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Usman, Muhammad
    Stockholm University, Faculty of Science, Department of Mathematics.
    RJ -Symmetric Laplace Operators on Star Graphs: Real Spectrum and Self-Adjointness2015In: Advances in Mathematical Physics, ISSN 1687-9120, E-ISSN 1687-9139, article id 649795Article in journal (Refereed)
    Abstract [en]

    How ideas of PJ -symmetric quantum mechanics can be applied to quantum graphs is analyzed, in particular to the star graph. The class of rotationally symmetric vertex conditions is analyzed. It is shown that all such conditions can effectively be described by circulantmatrices: real in the case of odd number of edges and complex having particular block structure in the even case. Spectral properties of the corresponding operators are discussed.

  • 3. Astudillo, Maria
    et al.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Usman, Muhammad
    Stockholm University, Faculty of Science, Department of Mathematics.
    RT-symmetric Laplace operators on star graphs: real spectrum and self-adjointness2015Report (Other academic)
    Abstract [en]

    In the current article it is analyzed how ideas of PT-symmetricquantum mechanics can be applied to quantum graphs, in particular tothe star graph. The class of rotationally-symmetric vertex conditionsis analyzed. It is shown that all such conditions can effectively be described bycirculant  matrices: real in the case of odd number of edges and complex having particular block structure  in the even case. Spectral properties of thecorresponding operators are discussed.

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  • 4. Avdonin, Sergei
    et al.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Inverse problems for quantum trees2008In: INVERSE PROBLEMS AND IMAGING, ISSN 1930-8337, Vol. 2, no 1, p. 1-21Article in journal (Refereed)
    Abstract [en]

    Three different inverse problems for the Schrodinger operator on a metric tree are considered, so far with standard boundary conditions at the vertices. These inverse problems are connected with the matrix Titchmarsh-Weyl function, response operator ( dynamic Dirichlet-to-Neumann map) and scattering matrix. Our approach is based on the boundary control ( BC) method and in particular on the study of the response operator. It is proven that the response operator determines the quantum tree completely, i.e. its connectivity, lengths of the edges and potentials on them. The same holds if the response operator is known for all but one boundary points, as well as for the Titchmarsh-Weyl function and scattering matrix. If the connectivity of the graph is known, then the lengths of the edges and the corresponding potentials are determined by just the diagonal terms of the data.

  • 5. Avdonin, Sergei
    et al.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Nowaczyk, Marlena
    INVERSE PROBLEMS FOR QUANTUM TREES II: RECOVERING MATCHING CONDITIONS FOR STAR GRAPHS2010In: Inverse Problems and Imaging, ISSN 1930-8337, Vol. 4, no 4, p. 579-598Article in journal (Refereed)
    Abstract [en]

    The inverse problem for the Schrodinger operator on a star graph is investigated. It is proven that such Schrodinger operator, i.e. the graph, the real potential on it and the matching conditions at the central vertex, can be reconstructed from the Titchmarsh-Weyl matrix function associated with the graph boundary. The reconstruction is also unique if the spectral data include not the whole Titchmarsh-Weyl function but its principal block (the matrix reduced by one dimension). The same result holds true if instead of the Titchmarsh-Weyl function the dynamical response operator or just its principal block is known.

  • 6. Bauch, S.
    et al.
    Ławniczak, M.
    Wrochna, J.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Sirko, L.
    Some Applications of Generalized Euler Characteristic of Quantum Graphs and Microwave Networks2021In: Acta Physica Polonica. A, ISSN 0587-4246, E-ISSN 1898-794X, Vol. 140, no 6, p. 525-531Article in journal (Refereed)
    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.

  • 7. Berkolaiko, Gregory
    et al.
    Kennedy, James B.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Mugnolo, Delio
    Edge connectivity and the spectral gap of combinatorial and quantum graphs2017In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 50, no 36, article id 365201Article in journal (Refereed)
    Abstract [en]

    We derive a number of upper and lower bounds for the first nontrivial eigenvalue of Laplacians on combinatorial and quantum graph in terms of the edge connectivity, i.e. the minimal number of edges which need to be removed to make the graph disconnected. On combinatorial graphs, one of the bounds corresponds to a well-known inequality of Fiedler, of which we give a new variational proof. On quantum graphs, the corresponding bound generalizes a recent result of Band and Levy. All proofs are general enough to yield corresponding estimates for the p-Laplacian and allow us to identify the minimizers. Based on the Betti number of the graph, we also derive upper and lower bounds on all eigenvalues which are 'asymptotically correct', i.e. agree with the Weyl asymptotics for the eigenvalues of the quantum graph. In particular, the lower bounds improve the bounds of Friedlander on any given graph for all but finitely many eigenvalues, while the upper bounds improve recent results of Ariturk. Our estimates are also used to derive bounds on the eigenvalues of the normalized Laplacian matrix that improve known bounds of spectral graph theory.

  • 8. Berkolaiko, Gregory
    et al.
    Kennedy, James B.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Mugnolo, Delio
    Impediments to diffusion in quantum graphs: Geometry-based upper bounds on the spectral gap2023In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 151, p. 3439-3455Article in journal (Refereed)
    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.

  • 9. Berkolaiko, Gregory
    et al.
    Kennedy, James B.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Mugnolo, Delio
    SURGERY PRINCIPLES FOR THE SPECTRAL ANALYSIS OF QUANTUM GRAPHS2019In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 372, no 7, p. 5153-5197Article in journal (Refereed)
    Abstract [en]

    We present a systematic collection of spectral surgery principles for the Laplacian on a compact metric graph with any of the usual vertex conditions (natural, Dirichlet, or delta-type) which show how various types of changes of a local or localised nature to a graph impact on the spectrum of the Laplacian. Many of these principles are entirely new; these include transplantation of volume within a graph based on the behaviour of its eigenfunctions, as well as unfolding of local cycles and pendants. In other cases we establish sharp generalisations, extensions, and refinements of known eigenvalue inequalities resulting from graph modification, such as vertex gluing, adjustment of vertex conditions, and introducing new pendant subgraphs. To illustrate our techniques we derive a new eigenvalue estimate which uses the size of the doubly connected part of a compact metric graph to estimate the lowest non-trivial eigenvalue of the Laplacian with natural vertex conditions. This quantitative isoperimetric-type inequality interpolates between two known estimates - one assuming the entire graph is doubly connected and the other making no connectivity assumption (and producing a weaker bound) - and includes them as special cases.

  • 10.
    Boman, Jan
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Suhr, Rune
    Stockholm University, Faculty of Science, Department of Mathematics.
    Schrödinger Operators on Graphs and Geometry II. Integrable Potentials and an Ambartsumian Theorem2016Report (Other academic)
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  • 11.
    Boman, Jan
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Suhr, Rune
    Stockholm University, Faculty of Science, Department of Mathematics.
    Schrödinger Operators on Graphs and Geometry II. Spectral Estimates for L-1-potentials and an Ambartsumian Theorem2018In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 90, no 3, article id 40Article in journal (Refereed)
    Abstract [en]

    In this paper we study Schrodinger operators with absolutely integrable potentials on metric graphs. Uniform bounds-i.e. depending only on the graph and the potential-on the difference between the eigenvalues of the Laplace and Schrodinger operators are obtained. This in turn allows us to prove an extension of the classical Ambartsumian Theorem which was originally proven for Schrodinger operators with Neumann conditions on an interval. We also extend a previous result relating the spectrum of a Schrodinger operator to the Euler characteristic of the underlying metric graph.

  • 12. Janas, Jan
    et al.
    Kurasov, PavelLund Institute of Technology, Sweden.Naboko, Sergei
    Spectral Methods for Operators of Mathematical Physics2004Collection (editor) (Refereed)
  • 13. Janas, Jan
    et al.
    Kurasov, PavelLund Institute of Technology, Sweden.Naboko, SergeiLaptev, AriStolz, Günter
    Methods of Spectral Analysis in Mathematical Physics: Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2006, Lund, Sweden2009Collection (editor) (Refereed)
  • 14. Janas, Jan
    et al.
    Kurasov, PavelStockholm University, Faculty of Science, Department of Mathematics.Naboko, SergeyLaptev, AriStolz, Gunter
    Spectral Theory and Analysis: Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2008, Bedlewo, Poland2011Conference proceedings (editor) (Refereed)
  • 15.
    Karreskog, Gustav
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Kupersmidt, I. Trygg
    Stockholm University, Faculty of Science, Department of Mathematics.
    SCHRODINGER OPERATORS ON GRAPHS: SYMMETRIZATION AND EULERIAN CYCLES2016In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 144, no 3, p. 1197-1207Article in journal (Refereed)
    Abstract [en]

    Spectral properties of the Schrodinger operator on a finite compact metric graph with delta-type vertex conditions are discussed. Explicit estimates for the lowest eigenvalue (ground state) are obtained using two different methods: Eulerian cycle and symmetrization techniques.

  • 16.
    Karreskog, Gustav
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trygg Kupersmidt, Isak
    Stockholm University, Faculty of Science, Department of Mathematics.
    Schrödinger operators on graphs: symmetrization and Eulerian cycles2015Report (Other academic)
    Abstract [en]

    Spectral properties of the Schrödinger operator on a finite compact metric graph with delta-type vertex conditions are discussed. Explicit estimates for the lowest eigenvalue (ground state) are obtained using two different methods:Eulerian cycle and symmetrization techniques. In the case of positive interactions even estimates for higher eigenvalues are derived.

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  • 17. Kennedy, James B.
    et al.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Léna, Corentin
    Mugnolo, Delio
    A theory of spectral partitions of metric graphs2021In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 60, no 2, article id 61Article in journal (Refereed)
    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.

  • 18. Kennedy, James B.
    et al.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Malenova, Gabriela
    Mugnolo, Delio
    On the Spectral Gap of a Quantum Graph2016In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 17, no 9, p. 2439-2473Article in journal (Refereed)
    Abstract [en]

    We consider the problem of finding universal bounds of isoperimetric or isodiametric type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature.

  • 19. Kennedy, James B.
    et al.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Malenová, Gabriela
    Mugnolo, Delio
    On the spectral gap of a quantum graph2015Report (Other academic)
    Abstract [en]

    We consider the problem of finding universal bounds of “isoperimetric” or “isodiametric” type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature.

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  • 20.
    Kiik, Jean-Claude
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Usman, Muhammad
    Stockholm University, Faculty of Science, Department of Mathematics.
    On vertex conditions for elastic systems2015Report (Other academic)
    Abstract [en]

    In this paper vertex conditions for the fourth order differential operator on the simplest metric graph - the $ Y$-graph, -are discussed. In order to make the operator symmetric one needs to impose extra conditions on the limit values offunctions and their derivatives at the central vertex. It is shown that such conditions corresponding to the free movement of beams depend on the angles between the beams in the equilibrium position.

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  • 21.
    Kiik, Jean-Claude
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Usman, Muhammad
    Stockholm University, Faculty of Science, Department of Mathematics.
    On vertex conditions for elastic systems2015In: Physics Letters A, ISSN 0375-9601, E-ISSN 1873-2429, Vol. 379, no 34-35, p. 1871-1876Article in journal (Refereed)
    Abstract [en]

    In this paper vertex conditions for the differential operator of fourth derivative on the simplest metric graph - the Y-graph, - are discussed. In order to make the operator symmetric one needs to impose extra conditions on the limit values of functions and their derivatives at the central vertex. It is shown that such conditions corresponding to the free movement of beams depend on the angles between the beams in the equilibrium position.

  • 22.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Always Detectable Eigenfunctions on Metric Graphs2021In: Acta Physica Polonica. A, ISSN 0587-4246, E-ISSN 1898-794X, Vol. 140, no 6, p. 510-513Article in journal (Refereed)
    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.

  • 23.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics. Lund University, Sweden; St. Petersburg University, Russia.
    Can one distinguish quantum trees from the boundary?2012In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 140, no 7, p. 2347-2356Article in journal (Refereed)
    Abstract [en]

    Schrödinger operators on metric trees are considered. It is proven that for certain matching conditions the Titchmarsh-Weyl matrix function does not determine the underlying metric tree; i.e. there exist quantum trees with equal Titchmarsh-Weyl functions. The constructed trees form one-parameter families of isospectral and isoscattering graphs.

  • 24.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Inverse problems for Aharonov-Bohm rings2010In: Mathematical proceedings of the Cambridge Philosophical Society (Print), ISSN 0305-0041, E-ISSN 1469-8064, Vol. 148, p. 331-362Article in journal (Refereed)
    Abstract [en]

    The inverse problem for Schrodinger operators on metric graphs is investigated in the presence of magnetic field. Graphs without loops and with Euler characteristic zero are considered. It is shown that the knowledge of the Tachmarsh-Weyl matrix function (Dirichletto-Neumann map) for just two values of the magnetic field allows one to reconstruct the graph and potential on it provided a certain additional no-resonance condition is satisfied.

  • 25.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Inverse Problems for Quantum Graphs: Recent Developments and Perspectives2011In: Acta Physica Polonica. A, ISSN 0587-4246, E-ISSN 1898-794X, Vol. 120, no 6A, p. A132-A141Article in journal (Refereed)
    Abstract [en]

    An introduction into the area of inverse problems for the Schrodinger operators on metric graphs is given. The case of metric finite trees is treated in detail with the focus on matching conditions. For graphs with loops we show that for almost all matching conditions the potential on the loop is not determined uniquely by the Titchmarsh-Weyl function. The class of all admissible potentials is characterized.

  • 26.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Inverse scattering for lasso graph2013In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 54, no 4, p. 042103-Article in journal (Refereed)
    Abstract [en]

    The inverse problem for the magnetic Schrodinger operator on the lasso graph with different matching conditions at the vertex is investigated. It is proven that the Titchmarsh-Weyl function known for different values of the magnetic flux through the cycle determines the unique potential on the loop, provided the entries of the vertex scattering matrix S parametrizing matching conditions satisfy s(12)s(23)s(31) not equal s(13)s(21)s(32). This is in contrast to numerous examples showing that the potential on the loop cannot be reconstructed from the boundary measurements.

  • 27.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    On Crossroads of Spectral Theory with Sergey Naboko2023In: 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.

  • 28.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    On the ground state for quantum graphs2019Report (Other academic)
    Abstract [en]

    The celebrated Perron-Frobenius is generalised for the case of quantum graphs -differential operators on metric graphs. It is shown that the theorem holds in the case of generalised delta couplings at the vertices - a new class ofvertex conditions introduced in the paper. Relations to positivity preserving and positivity improving semigroups are clarified.

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  • 29.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    On the ground state for quantum graphs2019In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 109, no 11, p. 2491-2512Article in journal (Refereed)
    Abstract [en]

    Ground-state eigenfunctions of Schrodinger operators can often be chosen positive. We analyse to which extent this is true for quantum graphs-differential operators on metric graphs. It is shown that the theorem holds in the case of generalised delta couplings at the vertices-a new class of vertex conditions introduced in the paper. It is shown that this class of vertex conditions is optimal. Relations to positivity preserving and positivity improving semigroups are clarified.

  • 30.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    On the Spectral Gap for Laplacians on Metric Graphs2013In: Acta Physica Polonica. A, ISSN 0587-4246, E-ISSN 1898-794X, Vol. 124, no 6, p. 1060-1062Article in journal (Refereed)
    Abstract [en]

    We discuss lower and upper estimates for the spectral gap of the Laplace operator on a finite compact connected metric graph. It is shown that the best lower estimate is given by the spectral gap for the interval with the same total length as the original graph. An explicit upper estimate is given by generalizing Cheeger's approach developed originally for Riemannian manifolds.

  • 31.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Spectral Gap for Complete Graphs: Upper and Lower Estimates2015Report (Other academic)
    Abstract [en]

    Lower and upper estimates for the spectral of the Laplacian on a compact metric graph are discussedà New upper estimates are presented and existing lower estimates are reviewedà The accuracy of these estimates is checked in the case of complete not necessarily regular graph with large number of vertices.

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  • 32.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Surgery of Graphs: M-Function and Spectral Gap2017In: Acta Physica Polonica. A, ISSN 0587-4246, E-ISSN 1898-794X, Vol. 132, no 6, p. 1666-1671Article in journal (Refereed)
    Abstract [en]

    We discuss behaviour of the spectral gap for quantum graphs when two metric graphs are glued together. It appears that precise answer to this question can be given using a natural generalisation of the Titchmarsh-Weyl M-functions.

  • 33.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Triplet extensions I: Semibounded operators in the scale of Hilbert spaces2009In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 107, p. 251-286Article in journal (Refereed)
    Abstract [en]

    The extension theory for semibounded symmetric operators is generalized by including operators acting in a triplet of Hilbert spaces. We concentrate our attention on the case where the minimal operator is essentially self-adjoint in the basic Hilbert space and construct a family of its self-adjoint extensions inside the triplet. All such extensions can be described by certain boundary conditions, and a natural counterpart of Krein's resolvent formula is obtained.

  • 34.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Understanding Quantum Graphs2019In: Acta Physica Polonica. A, ISSN 0587-4246, E-ISSN 1898-794X, Vol. 136, no 5, p. 797-802Article in journal (Refereed)
    Abstract [en]

    Current understanding of spectral asymptotics for quantum graphs is described. It is discussed how these results can be applied to inverse problems. In particular it is analysed whether the inverse spectral problem has a unique solution in the spirit of classical Ambartsumian theorem: a complete overview is provided.

  • 35.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Enerback, Magnus
    AHARONOV-BOHM RING TOUCHING A QUANTUM WIRE: HOW TO MODEL IT AND TO SOLVE THE INVERSE PROBLEM2011In: Reports on mathematical physics, ISSN 0034-4877, E-ISSN 1879-0674, Vol. 68, no 3, p. 271-287Article in journal (Refereed)
    Abstract [en]

    An explicitly solvable model of the gated Aharonov-Bohm ring touching a quantum wire is constructed and investigated. The inverse spectral and scattering problems are discussed. It is shown that the Titchmarsh-Weyl matrix function associated with the boundary vertices determines a unique electric potential on the graph even though the graph contains a loop. This system gives another family of isospectral quantum graphs.

  • 36.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Luger, Annemarie
    Stockholm University, Faculty of Science, Department of Mathematics.
    An Operator Theoretic Interpretation of the Generalized Titchmarsh-Weyl Coefficient for a Singular Sturm-Liouville Problem2011In: Mathematical physics, analysis and geometry, ISSN 1385-0172, E-ISSN 1572-9656, Vol. 14, no 2, p. 115-151Article in journal (Refereed)
    Abstract [en]

    In this article an operator theoretic interpretation of the generalized Titchmarsh-Weyl coefficient for the Hydrogen atom differential expression is given. As a consequence we obtain a new expansion theorem in terms of singular generalized eigenfunctions.

  • 37.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Luger, Annemarie
    Stockholm University, Faculty of Science, Department of Mathematics.
    Neuner, Christoph
    Stockholm University, Faculty of Science, Department of Mathematics.
    On supersingular perturbations of non-semibounded self-adjoint operators2019In: Journal of operator theory, ISSN 0379-4024, E-ISSN 1841-7744, Vol. 81, no 1, p. 195-223Article in journal (Refereed)
    Abstract [en]

    In this paper self-adjoint realizations of the formal expression A(alpha ):= A + alpha <phi, .> phi are described, where alpha is an element of R boolean OR {infinity}, the operator A is self-adjoint in a Hilbert space H and phi is a supersingular element from the scale space H--(n) (-2) (A) \H--(n) (-1) (A) for n >= 1. The crucial point is that the spectrum of A may consist of the whole real line. We construct two models to describe the family (A(alpha)). It can be interpreted in a Hilbert space with a twisted version of Krein's formula, or with a more classical version of Krein's formula but in a Pontryagin space. Finally, we compare the two approaches in terms of the respective Q-functions.

  • 38.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Luger, Annemarie
    Stockholm University, Faculty of Science, Department of Mathematics.
    Neuner, Christoph
    Stockholm University, Faculty of Science, Department of Mathematics.
    On supersingular perturbations of not necessarily semibounded self-adjoint operatorsManuscript (preprint) (Other academic)
  • 39.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lück, Alexander
    Mugnolo, Delio
    Wolf, Verena
    Stochastic hybrid models of gene regulatory networks - A PDE approach2018In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 305, p. 170-177Article in journal (Refereed)
    Abstract [en]

    A widely used approach to describe the dynamics of gene regulatory networks is based on the chemical master equation, which considers probability distributions over all possible combinations of molecular counts. The analysis of such models is extremely challenging due to their large discrete state space. We therefore propose a hybrid approximation approach based on a system of partial differential equations, where we assume a continuous-deterministic evolution for the protein counts. We discuss efficient analysis methods for both modeling approaches and compare their performance. We show that the hybrid approach yields accurate results for sufficiently large molecule counts, while reducing the computational effort from one ordinary differential equation for each state to one partial differential equation for each mode of the system. Furthermore, we give an analytical steady-state solution of the hybrid model for the case of a self-regulatory gene.

  • 40.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Majidzadeh Garjani, Babak
    Stockholm University, Faculty of Science, Department of Mathematics. Stockholm University, Faculty of Science, Department of Physics.
    Quantum Graphs: $ \mathcal{PT}$-symmetry and reflection symmetry of the spectrum2017Report (Other academic)
    Abstract [en]

    Not necessarily self-adjoint quantum graphs – differential operators on metric graphs – are considered. Assume in addition that the underlying metric graph possesses an automorphism (symmetry) $ \mathcal P $. If the differential operator is $ \mathcal P \mathcal T$-symmetric, then its spectrum has reflection symmetrywith respect to the real line. Our goal is to understand whether the opposite statement holds, namely whether the reflection symmetry of the spectrum ofa quantum graph implies that the underlying metric graph possesses a non-trivial automorphism and the differential operator is $ \mathcal P \mathcal T$-symmetric.We give partial answer to this question by considering equilateral star-graphs. The corresponding Laplace operator with Robin vertex conditions possesses reflection-symmetric spectrum if and only if the operator is $ \mathcal P \mathcal T$-symmetric with $ \mathcal P $ being an automorphism of the metric graph.

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  • 41.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Majidzadeh Garjani, Babak
    Stockholm University, Faculty of Science, Department of Physics.
    Quantum graphs: PT -symmetry and reflection symmetry of the spectrum2017In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 58, no 2, article id 023506Article in journal (Refereed)
    Abstract [en]

    Not necessarily self-adjoint quantum graphs-differential operators on metric graphs-are considered. Assume in addition that the underlying metric graph possesses an automorphism (symmetry) P. If the differential operator is PT -symmetric, then its spectrum has reflection symmetry with respect to the real line. Our goal is to understand whether the opposite statement holds, namely, whether the reflection symmetry of the spectrum of a quantum graph implies that the underlying metric graph possesses a non-trivial automorphism and the differential operator is PT symmetric. We give partial answer to this question by considering equilateral stargraphs. The corresponding Laplace operator with Robin vertex conditions possesses reflection-symmetric spectrum if and only if the operator is PT -symmetric with P being an automorphism of the metric graph.

  • 42.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Malenova, G.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Naboko, S.
    Spectral gap for quantum graphs and their edge connectivity2013In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 46, no 27, p. 275309-Article in journal (Refereed)
    Abstract [en]

    The spectral gap for Laplace operators on metric graphs and the relation between the graph's edge connectivity is investigated, in particular what happens to the gap if an edge is added to (or deleted from) a graph. It is shown that, in contrast to discrete graphs, the connection between the connectivity and the spectral gap is not one-to-one. The size of the spectral gap depends not only on the topology of the metric graph but on its geometric properties as well. It is shown that adding sufficiently large edges as well as cutting away sufficiently small edges leads to a decrease of the spectral gap. Corresponding explicit criteria are given.

  • 43.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Mugnolo, Delio
    Wolf, Verena
    Analytic solutions for stochastic hybrid models of gene regulatory networks2021In: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 82, no 1-2, article id 9Article in journal (Refereed)
    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.

  • 44.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Muller, Jacob
    Stockholm University, Faculty of Science, Department of Mathematics.
    Higher order differential operators on graphs and almost periodic functions2020Report (Other academic)
    Abstract [en]

    The spectra of -Laplacian operators  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.

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  • 45.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Muller, Jacob
    Stockholm University, Faculty of Science, Department of Mathematics.
    n-Laplacians on Metric Graphs and Almost Periodic Functions: I2021In: 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)
    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.

  • 46.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Muller, Jacob
    Stockholm University, Faculty of Science, Department of Mathematics.
    On the spectral gap for networks of beams2020Report (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.

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  • 47.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Naboko, Sergei
    Gluing graphs and the spectral gap: a Titchmarsh-Weyl matrix-valued function approach2020In: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 255, no 3, p. 303-326Article in journal (Refereed)
    Abstract [en]

    Assume that two metric graphs are joined by gluing a few vertices. We investigate the behaviour of the spectral gap for the corresponding standard Laplacians. It appears that a precise answer can be given in terms of the corresponding Titchmarsh-Weyl (matrix) functions of the two subgraphs, more precisely in terms of their negative spectral subspaces. We illustrate our results by considering explicit examples.

  • 48.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Naboko, Sergey
    Stockholm University, Faculty of Science, Department of Mathematics.
    GLUING GRAPHS AND SPECTRAL GAP: TITCHMARSH-WEYL OPERATOR-FUNCTION APPROACH2019Report (Other academic)
    Abstract [en]

    Assume that two metric graphs are joined together by gluing a few vertices. We investigate the behaviour of the spectral gap for the corresponding standard Laplacians. It appears that a precise answer can be given in terms of the corresponding Titchmarsh-Weyl (matrix) functions of the two subgraphs, more precisely in terms of their negative spectral subspaces. We illustrate our results by considering explicit examples.

    Download full text (pdf)
    fulltext
  • 49.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Naboko, Sergey
    Rayleigh estimates for differential operators on graphs2014In: Journal of Spectral Theory, ISSN 1664-039X, E-ISSN 1664-0403, Vol. 4, no 2, p. 211-219Article in journal (Refereed)
    Abstract [en]

    We study the spectral gap, i.e. the distance between the two lowest eigenvalues for Laplace operators on metric graphs. A universal lower estimate for the spectral gap is proven and it is shown that it is attained if the graph is formed by just one interval. Uniqueness of the minimizer allows to prove a geometric version of the Ambartsumian theorem derived originally for Schrodinger operators.

  • 50.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Naboko, Sergey
    Surgery of graphs and spectral gap: Titchmarsh-Weyl operator-function approach2017Report (Other academic)
    Abstract [en]

    Assume that two metric graphs are joined together by gluing fewv ertices. We investigate the behaviour of the spectral gap for the corresponding standard Laplacians. It appears that a precise answer can be given in terms of the corresponding Titchmarsh-Weyl (matrix) functions of the two subgraphs, more precisely in terms of their negative spectral subspaces. We illustrate our findings by considering explicit examples

    Download full text (pdf)
    fulltext
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