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Evolution Equations Associated with Self-Adjoint Operators

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Book cover Semigroup Methods for Evolution Equations on Networks

Part of the book series: Understanding Complex Systems ((UCS))

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Abstract

The theory of forms presented in Chap. 6 was originally developed in order to extend the study of parabolic problems beyond the setting of the Spectral Theorem, in much the same way the Lax–Milgram Lemma extended the applicability of the Riesz–Fréchet Theorem. This program was successful: Nowadays many relevant results on linear parabolic problems have been extended to the non-self-adjoint case by form methods and further results depend on much deeper techniques, including sophisticated functional calculi that conveniently replace the Spectral Theorem and whose exposition goes beyond the scope of our book, cf. [50].

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Notes

  1. 1.

    Here, GOE denotes the Gaussian orthogonal ensemble, the set of all symmetric matrices which becomes a measure space whenever it is endowed with a certain Gaussian measure. One similarly defines the ensembles of hermitian matrices GUE and of self-dual matrices GSE. We refer e.g. [57] for a good introduction to random matrix theory.

  2. 2.

    In fact reborn: the same name was earlier sporadically used in relation to Feynman diagrams.

References

  1. W. Arendt, Semigroups and evolution equations: Functional calculus, regularity and kernel estimates, in Handbook of Differential Equations: Evolutionary Equations – Vol. 1, ed. by C.M. Dafermos, E. Feireisl (North Holland, Amsterdam, 2004)

    Google Scholar 

  2. R. Band, T. Shapira, U. Smilansky, Nodal domains on isospectral quantum graphs: the resolution of isospectrality? J. Phys. A 39, 13999–14014 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. B. Bellazzini, M. Mintchev, Quantum fields on star graphs. J. Phys. A 39, 11101–11117 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. B. Bellazzini, M. Mintchev, P. Sorba, Bosonization and scale invariance on quantum wires. J. Phys. A 40, 2485–2507 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. J. von Below, A characteristic equation associated with an eigenvalue problem on c 2-networks. Lin. Algebra Appl. 71, 309–325 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  6. J. von Below, Can one hear the shape of a network? in Partial Differential Equations on Multistructures (Proc. Luminy 1999), ed. by F. Ali Mehmeti, J. von Below, S. Nicaise. Lecture Notes in Pure and Applied Mathematics, vol. 219 (Marcel Dekker, New York, 2001), pp. 19–36

    Google Scholar 

  7. J. von Below, J.A. Lubary, The eigenvalues of the Laplacian on locally finite networks. Result. Math. 47, 199–225 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. J. von Below, D. Mugnolo, The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions. Lin. Alg. Appl. 439, 1792–1814 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Berkolaiko, P. Kuchment, Introduction to Quantum Graphs. Mathematical Surveys and Monographs, vol. 186 (American Mathematical Society, Providence, 2013)

    Google Scholar 

  10. O. Bohigas, M.J. Giannoni, C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. 52, 1–4 (1984)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. J. Bolte, S. Endres, The trace formula for quantum graphs with general self-adjoint boundary conditions. Ann. Henri Poincaré A 10, 189–223 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. J. Bolte, J. Harrison, Spectral statistics for the Dirac operator on graphs. J. Phys. A 36, 2747–2769 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. D. Braess, Über ein Paradoxon aus der Verkehrsplanung. Unternehmensforschung 12, 258–268 (1968)

    MATH  MathSciNet  Google Scholar 

  14. J. Brüning, V. Geyler, K. Pankrashkin, Spectra of self-adjoint extensions and applications to solvable Schrödinger operators. Rev. Math. Phys. 20, 1–70 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. S. Butler, Interlacing for weighted graphs using the normalized Laplacian. Electronic J. Lin. Alg. 16, 87 (2007)

    Google Scholar 

  16. S.K. Butler, Eigenvalues and Structures of Graphs. PhD thesis, University of California, San Diego, 2008

    Google Scholar 

  17. R. Carlson, Eigenvalue cluster traces for quantum graphs with equal edge lengths. Rocky Mount. J. Math. 42, 467–490 (2012)

    Article  MATH  Google Scholar 

  18. C. Cattaneo, The spectrum of the continuous Laplacian on a graph. Monats. Math. 124, 124–215 (1997)

    Article  MathSciNet  Google Scholar 

  19. C. Cattaneo, The spread of the potential on a homogeneous tree. Ann. Mat. Pura Appl., IV Ser. 175, 29–57 (1998)

    Google Scholar 

  20. C. Cattaneo, The spread of the potential on a weighted graph. Rend. Semin. Mat. Torino 57, 221–229 (1999)

    MATH  MathSciNet  Google Scholar 

  21. C. Cattaneo, L. Fontana, D’Alembert formula on finite one-dimensional networks. J. Math. Anal. Appl. 284, 403–424 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. G. Chen, G. Davis, F. Hall, Z. Li, K. Patel, M. Stewart, An interlacing result on normalized Laplacians. SIAM J. Discr. Math. 18, 353–361 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. R. Chill, V. Keyantuo, M. Warma, Generation of cosine families on L p(0, 1) by elliptic operators with Robin boundary conditions, in Functional Analysis and Evolution Equations, ed.by H. Amann et al. (Birkhäuser, Basel, 2008), pp. 113–130

    Google Scholar 

  24. F.R.K. Chung, Spectral Graph Theory. Regional Conference Series in Mathematics, vol. 92 (American Mathematical Society, Providence, 1997)

    Google Scholar 

  25. Y. Colin de Verdière, Spectre du laplacien et longueurs des géodésiques périodiques. ii. Compos. Math. 27, 159–184 (1973)

    Google Scholar 

  26. T. Coulhon, Heat kernel estimates, sobolev-type inequalities and riesz transform on noncompact riemannian manifolds, in Analysis and geometry of metric measure spaces (Proc. Montréal 2011). CRM Proceedings and Lecture Notes, vol. 56 (American Mathematical Society, Providence, 2013), pp. 55–66

    Google Scholar 

  27. T. Coulhon, A. Sikora, Gaussian heat kernel upper bounds via the Phragmén–Lindelöf theorem. Proc. London Math. Soc. 96, 507–544 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. R. Dáger, E. Zuazua, Wave propagation, observation and control in 1-d flexible multi-structures. Mathématiques et Applications, vol. 50 (Springer, Berlin, 2005)

    Google Scholar 

  29. E.B. Davies, Heat kernel bounds, conservation of probability and the Feller property. J. Anal. Math. 58, 99–119 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  30. E.B. Davies, Large deviations for heat kernels on graphs. J. London Math. Soc. 2, 65–72 (1993)

    Article  Google Scholar 

  31. T. Delmotte, Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoam. 15, 181–232 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  32. R.J. Duffin, Equipartition of energy in wave motion. J. Math. Anal. Appl. 32, 386–391 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  33. K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194 (Springer, New York, 2000)

    Google Scholar 

  34. S.A. Fulling, P. Kuchment, J.H. Wilson, Index theorems for quantum graphs. J. Phys. A 40, 14165–14180 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  35. S. Gnutzmann, U. Smilansky, Quantum graphs: Applications to quantum chaos and universal spectral statistics. Adv. Phys 55, 527–625 (2006)

    Article  ADS  Google Scholar 

  36. J.A. Goldstein, An asymptotic property of solutions of wave equations. Proc. Am. Math. Soc. 23, 359–363 (1969)

    Article  MATH  Google Scholar 

  37. A. Grigor’yan, A. Telcs, Sub-gaussian estimates of heat kernels on infinite graphs. Duke Math. J. 109, 451–510 (2001)

    Google Scholar 

  38. B. Gutkin, U. Smilansky, Can one hear the shape of a graph? J. Phys. A 34, 6061–6068 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  39. P.R. Halmos, What does the spectral theorem say? Am. Math. Mon 70, 241–247 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  40. D. Horak, J. Jost, Interlacing inequalities for eigenvalues of discrete Laplace operators. Ann. Global Anal. Geom. 43, 177–207 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  41. M. Kac, Can one hear the shape of a drum? Am. Math. Mon 73, 1–23 (1966)

    Article  MATH  Google Scholar 

  42. B. Klöss, The flow approach for waves in networks. Oper. Matrices 6, 107–128 (2012)

    Article  MathSciNet  Google Scholar 

  43. A.V. Kopytin, On representation of solutions to the wave equation on graphs with commensurable edges. Tr. Mat. Fak. Voronezh. Gos. Univ. 6, 67–77 (2001)

    Google Scholar 

  44. V. Kostrykin, J. Potthoff, R. Schrader, Heat kernels on metric graphs and a trace formula, in Adventures in Mathematical Physics. Contemporary Mathematics, vol. 447 (American Mathematical Society, Providence, 2007), pp. 175–198

    Google Scholar 

  45. T. Kottos, U. Smilansky, Quantum chaos on graphs. Phys. Rev. Lett. 79, 4794–4797 (1997)

    Article  ADS  Google Scholar 

  46. T. Kottos, U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs. Ann. Physics 274, 76–124 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  47. M. Kramar Fijavž, D. Mugnolo, E. Sikolya, Variational and semigroup methods for waves and diffusion in networks. Appl. Math. Optim. 55, 219–240 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  48. P. Kuchment, Quantum graphs I: Some basic structures. Waves Random Media 14, 107–128 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  49. P. Kuchment, Quantum graphs: an introduction and a brief survey, in Analysis on Graphs and its Applications, ed. by P. Exner, J. Keating, P. Kuchment, T. Sunada, A. Teplyaev. Proceedings of Symposia in Pure Mathematics, vol. 77 (American Mathematical Society, Providence, 2008), pp. 291–314

    Google Scholar 

  50. P.C. Kunstmann, L. Weis, Maximal L p -regularity for parabolic equations, Fourier multiplier theorems and H ∞-functional calculus, in Functional Analytic Methods for Evolution Equations. Lecture Notes in Mathematics, vol. 1855 (Springer, Berlin, 2004), pp. 65–311

    Google Scholar 

  51. P. Kurasov, G. Malenová, S. Naboko, Spectral gap for quantum graphs and their connectivity. J. Phys. A 46, 275309 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  52. P. Kurasov, M. Nowaczyk, Inverse spectral problem for quantum graphs. J. Phys. A 38, 4901 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  53. P. Kurasov, A. Posilicano, Finite speed of propagation and local boundary conditions for wave equations with point interactions. Proc. Am. Math. Soc. 133, 3071–3078 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  54. J.E. Lagnese, G. Leugering, E.J.P.G. Schmidt, Modeling, Analysis, and Control of Dynamic Elastic Multi-Link Structures. Systems and Control: Foundations and Applications (Birkhäuser, Basel, 1994)

    Book  MATH  Google Scholar 

  55. D. Lenz, K. Pankrashkin, New relations between discrete and continuous transition operators on (metric) graphs. arXiv:1305.7491

    Google Scholar 

  56. G. Malenová, Spectra of quantum graphs. Master’s thesis, Czech Technical University in Prague, 2013

    Google Scholar 

  57. M.L. Mehta, Random Matrices. Pure and Applied Mathematics, vol. 142 (Elsevier, Amsterdam, 2004)

    Google Scholar 

  58. B. Mohar, The Laplacian spectrum of graphs. Graph Theory Comb Appl 2, 871–898 (1991)

    MathSciNet  Google Scholar 

  59. D. Mugnolo, Gaussian estimates for a heat equation on a network. Netw. Het. Media 2, 55–79 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  60. S. Nicaise, Approche spectrale des problemes de diffusion sur les réseaux, in Séminaire de Théorie du Potentiel Paris, vol. 8 (Springer, Berlin, 1987), pp. 120–140

    Google Scholar 

  61. S. Nicaise, Spectre des réseaux topologiques finis. Bull. Sci. Math. II. Sér. 111, 401–413 (1987)

    MATH  MathSciNet  Google Scholar 

  62. M.G. Pala, S. Baltazar, P. Liu, H. Sellier, B. Hackens, F. Martins, V. Bayot, X. Wallart, L. Desplanque, S. Huant, Transport inefficiency in branched-out mesoscopic networks: An analog of the Braess paradox. Phys. Rev. Lett. 108, 076802 (2012)

    Article  ADS  Google Scholar 

  63. M.M.H. Pang, The heat kernel of the Laplacian defined on a uniform grid. Semigroup Forum 78, 238–252 (2008)

    Article  Google Scholar 

  64. K. Pankrashkin, Spectra of Schrödinger operators on equilateral quantum graphs. Lett. Math. Phys. 77, 139–154 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  65. O. Post, First order approach and index theorems for discrete and metric graphs. Ann. Henri Poincaré A 10, 823–866 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  66. R. Pröpper, Heat kernel bounds for the Laplacian on metric graphs of polygonal tilings. Semigroup Forum 86, 262–271 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  67. V. Pryadiev, Description of the solution of an initial-boundary value problem for the wave equation on a one-dimensional spatial network in terms of the green function of the corresponding boundary value problem for an ordinary differential equation. J. Math. Sci. 147, 6470–6482 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  68. J.-P. Roth, Spectre du laplacien sur un graphe. C. R. Acad. Sci. Paris Sér. I Math. 296, 793–795 (1983)

    Google Scholar 

  69. J.-P. Roth, Le spectre du laplacien sur un graphe, in Colloque de Théorie du Potentiel - Jacques Deny (Proc. Orsay 1983), ed. by G. Mokobodzki, D. Pinchon. Lecture Notes in Mathematics, vol. 1096 (Springer, Berlin, 1984), pp. 521–539

    Google Scholar 

  70. R. Schrader, Finite propagation speed and free quantum fields on networks. J. Phys. A 42, 495401 (2009)

    Article  MathSciNet  Google Scholar 

  71. A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20, 47–87 (1956)

    MATH  MathSciNet  Google Scholar 

  72. A. Sikora, Riesz transform, Gaussian bounds and the method of wave equation. Math. Z. 247, 643–662 (2004)

    MATH  MathSciNet  Google Scholar 

  73. M.H. Stone, On one-parameter unitary groups in Hilbert space. Ann. Math. 33, 643–648 (1932)

    Article  Google Scholar 

  74. K.-T. Stur, Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32, 275–312 (1995)

    Google Scholar 

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  1. Universität Ulm Institut für Analysis, Ulm, Germany

    Delio Mugnolo

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Mugnolo, D. (2014). Evolution Equations Associated with Self-Adjoint Operators. In: Semigroup Methods for Evolution Equations on Networks. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-04621-1_7

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