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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.

  • 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. 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.

  • 7.
    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)
  • 8.
    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.

  • 9. Janas, Jan
    et al.
    Kurasov, PavelLund Institute of Technology, Sweden.Naboko, Sergei
    Spectral Methods for Operators of Mathematical Physics2004Collection (editor) (Refereed)
  • 10. 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)
  • 11. 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)
  • 12.
    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.

  • 13. 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.

  • 14.
    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.

  • 15.
    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.

  • 16.
    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.

  • 17.
    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.

  • 18.
    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.

  • 19.
    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.

  • 20.
    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.

  • 21.
    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.

  • 22.
    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.

  • 23.
    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.

  • 24.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Karreskog, Gustav
    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.

  • 25.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Kennedy, James B.
    Universität Stuttgart.
    Malenova, Gabriela
    Kungliga Tekniska Högskolan, Stockholm.
    Mugnolo, Delio
    FernUniversität Hagen.
    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.

  • 26.
    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.

  • 27.
    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.

  • 28.
    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)
  • 29.
    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.

  • 30.
    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.

  • 31.
    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.

  • 32.
    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.

  • 33.
    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.

  • 34.
    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.

  • 35.
    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)
  • 36.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Ogik, Rao
    Stockholm University, Faculty of Science, Department of Mathematics.
    ON EQUI-TRANSMITTING MATRICES2016In: Reports on mathematical physics, ISSN 0034-4877, E-ISSN 1879-0674, Vol. 78, no 2, p. 199-218Article in journal (Refereed)
    Abstract [en]

    Equi-transmitting scattering matrices are studied. A complete description of such matrices up to order five is given. It is shown that the standard matching conditions matrix is essentially the only equi-transmitting matrix for orders 3 and 5. For orders 4 and 6, there exists other equi-transmitting ones but all such matrices have zero trace.

  • 37.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Ogik, Rao
    Stockholm University, Faculty of Science, Department of Mathematics.
    On Equi-transmitting matrices2014Report (Other academic)
    Abstract [en]

    Equi-transmitting scattering matrices are studied. A complete description of such matrices up to order six is given. It is shown that the standard matching conditions matrix is essentially the only equi-transmitting matrix for orders 3 and 5. For orders 4 and 6, there exists other equi-transmitting but all such matrices have zero trace.

  • 38.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Serio, Andrea
    Stockholm University, Faculty of Science, Department of Mathematics.
    On the Sharpness of Spectral Estimates for Graph Laplacians2018In: Reports on mathematical physics, ISSN 0034-4877, E-ISSN 1879-0674, Vol. 82, no 1, p. 63-80Article in journal (Refereed)
    Abstract [en]

    We prove that the upper spectral estimate for quantum graphs due to Berkolaiko-Kennedy-Kurasov-Mugnolo [5] is sharp.

  • 39.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Serio, Andrea
    Stockholm University, Faculty of Science, Department of Mathematics.
    Optimal Potentials for Quantum Graphs2019In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 20, no 5, p. 1517-1542Article in journal (Refereed)
    Abstract [en]

    Schrödinger operators on metric graphs with delta couplings at the vertices are studied. We discuss which potential and which distribution of delta couplings on a given graph maximise the ground state energy, provided the integral of the potential and the sum of strengths of the delta couplings are fixed. It appears that the optimal potential if it exists is a constant function on its support formed by a set of intervals separated from the vertices. In the case where the optimal configuration does not exist explicit optimising sequences are presented.

  • 40.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Serio, Andrea
    Stockholm University, Faculty of Science, Department of Mathematics.
    Topological damping of Aharonov-Bohm effect: quantum graphs and vertex conditions2015Report (Other academic)
  • 41.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Simonov, Sergey
    Weyl-Titchmarsh-type formula for periodic Schrodinger operator with Wigner-von Neumann potential2013In: Proceedings of the Royal Society of Edinburgh. Section A Mathematics, ISSN 0308-2105, E-ISSN 1473-7124, Vol. 143, no 2, p. 401-425Article in journal (Refereed)
    Abstract [en]

    The Schrodinger operator on the half-line with periodic background potential perturbed by a certain potential of Wigner-von Neumann type is considered. The asymptotics of generalized eigenvectors for lambda is an element of C+ and on the absolutely continuous spectrum is established. The Weyl-Titchmarsh-type formula for this operator is proven.

  • 42.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Suhr, Rune
    Stockholm University, Faculty of Science, Department of Mathematics.
    ASYMPTOTICALLY ISOSPECTRAL QUANTUM GRAPHS AND TRIGONOMETRIC POLYNOMIALS2018Report (Other academic)
    Abstract [en]

    The theory of almost periodic functions is used to investigate spectral properties of Schrödinger operators on metric graphs, also known as quantum graphs. In particular we prove that two Schrödinger 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 Schrödinger operator is isospectral to the standard Laplacian on a may be different metric graph only if the potential is identically equal to zero.

  • 43.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Suhr, Rune
    Stockholm University, Faculty of Science, Department of Mathematics.
    SCHRÖDINGER OPERATORS ON GRAPHS AND GEOMETRY III: GENERAL VERTEX CONDITIONS AND COUNTEREXAMPLES2018Report (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 reference Laplacians. A geometric version of Ambartsumian theorem is proven under the assumptionthat 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.

  • 44.
    Kurasov, Pavel
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Suhr, Rune
    Stockholm University, Faculty of Science, Department of Mathematics.
    Schrödinger operators on graphs and geometry. III. General vertex conditions and counterexamples2018In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 59, no 10, article id 102104Article in journal (Refereed)
    Abstract [en]

    Schrödinger operators on metric graphs with general vertex conditions are studied. Explicit spectral asymptotics is derived in terms of the spectrum of reference Laplacians. A geometric version of the 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 Ambartsumian theorem does not hold in general without additional assumptions on the vertex conditions.

  • 45.
    Suhr, Rune
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Asymptotically isospectral quantum graphs and trigonometric polynomialsManuscript (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.

  • 46.
    Suhr, Rune
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Kurasov, Pavel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Schrödinger operators on graphs: a geometric version of the Ambartsumian theoremManuscript (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.

1 - 46 of 46
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