Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T17:16:27.251Z Has data issue: false hasContentIssue false
  • English
  • Français

Harmonic Analysis on Metrized Graphs

Published online by Cambridge University Press:  20 November 2018

Matt Baker  and
Robert Rumely
Matt Baker
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, U.S.A. email: mbaker@math.gatech.edu
Robert Rumely
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A. email: rr@math.uga.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper studies the Laplacian operator on a metrized graph, and its spectral theory.

Keywords

43A9958C4005C99metrized graphharmonic analysiseigenfunction
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 2 , 01 April 2007 , pp. 225 - 275
Copyright
Copyright © Canadian Mathematical Society 2007

References

[AM1] Ali Mehmeti, F., A characterization of a generalized C∞-notion on nets. Integral Equations Operator Theory 9(1986), no. 6, 753–766.Google Scholar
[AM2] Ali Mehmeti, F., Regular solutions of transmission and interaction problems for wave equations. Math. Methods Appl. Sci. 11(1989), no. 5, 665–685.Google Scholar
[AM3] Ali Mehmeti, F., Nonlinear Waves in Networks. Mathematical Research 80, Akademie-Verlag, Berlin, 1994.Google Scholar
[BF] Baker, M. and Faber, X., Metrized graphs and electrical networks. In: Quantum Graphs and Their Applications, Contemporary Mathematics 415, American Mathematical Society, Providence, RI, 2006.Google Scholar
[BR] Baker, M. and Rumely, R., Equidistribution of small points, rational dynamics, and potential theory. Ann. Inst. Fourier (Grenoble) 56(2006), no. 3, 625–688.Google Scholar
[Be1] von Below, J., A characteristic equation associated to an eigenvalue problem on c 2-networks. Linear Algebra Appl. 71(1985), 309–325.Google Scholar
[Be2] von Below, J., Sturm-Liouville eigenvalue problems on networks. Math. Methods Appl. Sci. 10(1988), no. 4, 383–395.Google Scholar
[Berk] Berkovich, V. G., Spectral Theory and Analytic Geometry over Non-Archimedean Fields. Mathematical Surveys and Monographs 33, American Mathematical Society, Providence, RI, 1990.Google Scholar
[PE] Bassanini, P. and Elcrat, A., Theory and Applications of Partial Differential Equations. Mathematical Concepts and Methods in Science and Engineering 46. Plenum Press, New York, 1997.Google Scholar
[Bi] Biggs, N., Algebraic Graph Theory. Second edition. Cambridge University Press, Cambridge, 1993.Google Scholar
[Ca] Cattaneo, C., The spectrum of the continuous Laplacian on a graph. Monatsh. Math. 124(1997), no. 3, 215–235.Google Scholar
[CR] Chinburg, T. and Rumely, R., The capacity pairing. J. Reine Angew. Math. 434(1993), 1–44.Google Scholar
[Ch] Chung, F., Spectral Graph Theory. CBMS Regional Conference Series in Mathematics 92. American Mathematical Society, Providence, RI, 1997.Google Scholar
[Fa] Faber, X., Spectral convergence of the discrete Laplacian on models of a metrized graph. New York J. Math. 12(2006), 97–121.Google Scholar
[Fal] Faltings, G., Calculus on arithmetic surfaces,. Ann. of Math. 119(1984), no. 2, 387–424.Google Scholar
[FJ] Favre, C. and Jonsson, M., The Valuative Tree. Lecture Notes in Mathematics 1853. Springer-Verlag, Berlin, 2004.Google Scholar
[Fr] Friedlander, L., Genericity of simple eigenvalues for a metric graph. Israel J. Math. 146(2005), 149–156.Google Scholar
[Fr2] Friedlander, L., Extremal properties of eigenvalues for a metric graph. Ann. Inst. Fourier (Grenoble) 55(2005), no. 1, 199–211.Google Scholar
[GO] Gaveau, B., Okada, M., and Okada, T., Explicit heat kernels on graphs and spectral analysis. In: Several Complex Variables, Math. Notes 38, Princeton University Press, Princeton, NJ, 1991, pp. 364–388.Google Scholar
[GR] Godsil, C. and Royle, G., Algebraic Graph Theory. Graduate Texts in Mathematics 207. Springer-Verlag, New York, 2001.Google Scholar
[HS] Hindry, M and Silverman, J., On Lehmer's conjecture for elliptic curves. In: Séminaire de Théorie des Nombres. Progr. Math. 91. Birkhäuser Boston, Boston, MA, 1990, pp. 103–116.Google Scholar
[Ki] Kigami, J., Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal. 128(1995), no. 1, 48–86.Google Scholar
[Ku] Kuchment, P., Quantum graphs: I. Some basic structures. Waves Random Media. 14(2004), no. 1, S107–S128.Google Scholar
[La] Lang, S., Real Analysis. Addison-Wesley, Reading, 1969.Google Scholar
[Mo] Mohar, B., Some applications of Laplace eigenvalues of graphs. In: Graph Symmetry: Algebraic Methods and Applications. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 497, Kluwer, Dordrecht, 1997, pp. 225–275.Google Scholar
[N1] Nicaise, S., Some results on spectral theory over networks, applied to nerve impulse transmission. In: Orthogonal Polynomials and Applications. Lecture Notes in Mathematics 1171. Springer-Verlag, Berlin, 1985, pp. 532–541.Google Scholar
[N2] Nicaise, S., Approche spectrale des problèmes de diffusion sur les résaux. In: Séminaire de Thórie du Potentiel. Lecture Notes in Mathematics 1235. Springer-Verlag, Berlin, 1989, pp. 120–140.Google Scholar
[Ok] Okada, T., Asymptotic behavior of skew conditional heat kernels on graph networks. Canad. J. Math 45(1993), no. 4, 863–878.Google Scholar
[RN] Riesz, F. and Sz.-Nagy, B., Functional Analysis. Ungar, New York, 1955.Google Scholar
[Ro] Roth, J. P., Le spectra du laplacien sur un graphe. In: Théorie du potentiel. Lecture Notes in Mathematics 1096. Springer-Verlag, Berlin, 1984, pp. 521–538.Google Scholar
[Roy] Royden, H. L., Real Analysis. Third edition. MacMillan, New York, 1998.Google Scholar
[Rud] Rudin, W., Principles of Mathematical Analysis. Third edition. McGraw-Hill, New York, 1976.Google Scholar
[Ru] Rumely, R., Capacity Theory on Algebraic Curves. Lecture Notes in Mathematics 1378. Springer-Verlag, Berlin, 1989.Google Scholar
[RB] Rumely, R. and Baker, M., Analysis and dynamics on the Berkovich projective line. preprint available at http://arxiv.org/abs/math.NT/0407433. Google Scholar
[Si] Simmons, G., Differential Equations with Applications and Historical Notes. McGraw-Hill, New York, 1972.Google Scholar
[So] Solomyak, M., On the spectrum of the Laplacian on regular metric trees. Waves Random Media. 14(2004), no. 1, S155–S171.Google Scholar
[Zh] Zhang, S., Admissible pairing on a curve. Invent.Math. 112(1993), no. 1, 171–193.Google Scholar