# American Institute of Mathematical Sciences

June  2013, 6(3): 783-791. doi: 10.3934/dcdss.2013.6.783

## Dispersive waves with multiple tunnel effect on a star-shaped network

 1 Université de Valenciennes et du Hainaut-Cambrésis, LAMAV, FR CNRS 2956, F-59313 Valenciennes, France 2 TU Darmstadt, Fachbereich Mathematik, Schloßgartenstraße 7, D-64289 Darmstadt, Germany, Germany

Received  April 2010 Revised  December 2010 Published  December 2012

We consider the Klein-Gordon equation on a star-shaped network composed of $n$ half-axes connected at their origins. We add a potential which is constant but different on each branch. The corresponding spatial operator is self-adjoint and we state explicit expressions for its resolvent and its resolution of the identity in terms of generalized eigenfunctions. This leads to a generalized Fourier type inversion formula in terms of an expansion in generalized eigenfunctions. This paper is a survey of a longer article, nevertheless the proof of the central formula is indicated.
Citation: F. Ali Mehmeti, R. Haller-Dintelmann, V. Régnier. Dispersive waves with multiple tunnel effect on a star-shaped network. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 783-791. doi: 10.3934/dcdss.2013.6.783
##### References:
 [1] F. Ali Mehmeti, Spectral theory and $L^{\infty}$-time decay estimates for Klein-Gordon equations on two half axes with transmission: The tunnel effect, Math. Methods Appl. Sci., 17 (1994), 697-752. doi: 10.1002/mma.1670170904. [2] F. Ali Mehmeti, "Transient Tunnel Effect and Sommerfeld Problem: Waves in Semi-Infinite Structures," Mathematical Research, 91, Akademie Verlag, Berlin, 1996. [3] F. Ali Mehmeti, R. Haller-Dintelmann and V. Régnier, Expansions in generalized eigenfunctions of the weighted Laplacian on star-shaped networks, in "Functional Analysis and Evolution Equations: The Günter Lumer Volume" (eds. H. Amann, W. Arendt, M. Hieber, F. Neubrander, S. Nicaise and J. von Below), Birkhäuser, Basel, (2007), 1-16. doi: 10.1007/978-3-7643-7794-6_1. [4] F. Ali Mehmeti, R. Haller-Dintelmann and V. Régnier, Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation, J. Evol. Equ., 12 (2012), 513-545 arXiv:1012.3068v1. doi: 10.1007/s00028-012-0143-5. [5] F. Ali Mehmeti and V. Régnier, Splitting of energy of dispersive waves in a star-shaped network, Z. Angew. Math. Mech., 83 (2003), 105-118. doi: 10.1002/zamm.200310010. [6] F. Ali Mehmeti and V. Régnier, Delayed reflection of the energy flow at a potential step for dispersive wave packets, Math. Methods Appl. Sci., 27 (2004), 1145-1195. doi: 10.1002/mma.484. [7] F. Ali Mehmeti and V. Régnier, Global existence and causality for a transmission problem with a repulsive nonlinearity, Nonlinear Anal., 69 (2008), 408-424. doi: 10.1016/j.na.2007.05.028. [8] J. von Below and J. A. Lubary, The eigenvalues of the Laplacian on locally finite networks Results Math., 47 (2005), 199-225. [9] S. Cardanobile and D. Mugnolo, Parabolic systems with coupled boundary conditions, J. Differential Equations, 247 (2009), 1229-1248. doi: 10.1016/j.jde.2009.04.013. [10] Y. Daikh, "Temps de Passage de Paquets D'ondes de Basses Fréquences ou Limités en Bandes de Fréquences par une Barrière de Potentiel," Thèse de Doctorat, Université de Valenciennes, France, 2004. [11] J. M. Deutch and F. E. Low, Barrier penetration and superluminal velocity, Annals of Physics, 228 (1993), 184-202. doi: 10.1006/aphy.1993.1092. [12] N. Dunford and J. T. Schwartz, "Linear Operators II," Wiley Interscience, New York, 1963. [13] A. Enders and G. Nimtz, On superluminal barrier traversal, J. Phys. I France, 2 (1992), 1693-1698. [14] A. Haibel and G. Nimtz, Universal relationship of time and frequency in photonic tunnelling, Ann. Physik (Leipzig), 10 (2001), 707-712. [15] V. Kostrykin and R. Schrader, The inverse scattering problem for metric graphs and the travelling salesman problem,, preprint, (). [16] M. Pozar, "Microwave Engineering," Addison-Wesley, New York, 1990. [17] J. Weidmann, "Spectral Theory of Ordinary Differential Operators," Lecture Notes in Mathematics, 1258, Springer-Verlag, Berlin, 1987.

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##### References:
 [1] F. Ali Mehmeti, Spectral theory and $L^{\infty}$-time decay estimates for Klein-Gordon equations on two half axes with transmission: The tunnel effect, Math. Methods Appl. Sci., 17 (1994), 697-752. doi: 10.1002/mma.1670170904. [2] F. Ali Mehmeti, "Transient Tunnel Effect and Sommerfeld Problem: Waves in Semi-Infinite Structures," Mathematical Research, 91, Akademie Verlag, Berlin, 1996. [3] F. Ali Mehmeti, R. Haller-Dintelmann and V. Régnier, Expansions in generalized eigenfunctions of the weighted Laplacian on star-shaped networks, in "Functional Analysis and Evolution Equations: The Günter Lumer Volume" (eds. H. Amann, W. Arendt, M. Hieber, F. Neubrander, S. Nicaise and J. von Below), Birkhäuser, Basel, (2007), 1-16. doi: 10.1007/978-3-7643-7794-6_1. [4] F. Ali Mehmeti, R. Haller-Dintelmann and V. Régnier, Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation, J. Evol. Equ., 12 (2012), 513-545 arXiv:1012.3068v1. doi: 10.1007/s00028-012-0143-5. [5] F. Ali Mehmeti and V. Régnier, Splitting of energy of dispersive waves in a star-shaped network, Z. Angew. Math. Mech., 83 (2003), 105-118. doi: 10.1002/zamm.200310010. [6] F. Ali Mehmeti and V. Régnier, Delayed reflection of the energy flow at a potential step for dispersive wave packets, Math. Methods Appl. Sci., 27 (2004), 1145-1195. doi: 10.1002/mma.484. [7] F. Ali Mehmeti and V. Régnier, Global existence and causality for a transmission problem with a repulsive nonlinearity, Nonlinear Anal., 69 (2008), 408-424. doi: 10.1016/j.na.2007.05.028. [8] J. von Below and J. A. Lubary, The eigenvalues of the Laplacian on locally finite networks Results Math., 47 (2005), 199-225. [9] S. Cardanobile and D. Mugnolo, Parabolic systems with coupled boundary conditions, J. Differential Equations, 247 (2009), 1229-1248. doi: 10.1016/j.jde.2009.04.013. [10] Y. Daikh, "Temps de Passage de Paquets D'ondes de Basses Fréquences ou Limités en Bandes de Fréquences par une Barrière de Potentiel," Thèse de Doctorat, Université de Valenciennes, France, 2004. [11] J. M. Deutch and F. E. Low, Barrier penetration and superluminal velocity, Annals of Physics, 228 (1993), 184-202. doi: 10.1006/aphy.1993.1092. [12] N. Dunford and J. T. Schwartz, "Linear Operators II," Wiley Interscience, New York, 1963. [13] A. Enders and G. Nimtz, On superluminal barrier traversal, J. Phys. I France, 2 (1992), 1693-1698. [14] A. Haibel and G. Nimtz, Universal relationship of time and frequency in photonic tunnelling, Ann. Physik (Leipzig), 10 (2001), 707-712. [15] V. Kostrykin and R. Schrader, The inverse scattering problem for metric graphs and the travelling salesman problem,, preprint, (). [16] M. Pozar, "Microwave Engineering," Addison-Wesley, New York, 1990. [17] J. Weidmann, "Spectral Theory of Ordinary Differential Operators," Lecture Notes in Mathematics, 1258, Springer-Verlag, Berlin, 1987.
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