Abstract
We consider the Klein-Gordon equation on a star-shaped network composed of n half-axes connected at their origins.W e add a potential that is constant but different on each branch.E xploiting a spectral theoretic solution formula from a previous paper, we study the L∞ -time decay via Hörmander’s version of the stationary phase method.W e analyze the coefficient c of the leading term \(c\cdot t^{ - {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace}2}}\) of the asymptotic expansion of the solution with respect to time.F or two branches we prove that for an initial condition in an energy band above the threshold of tunnel effect, this coefficient tends to zero on the branch with the higher potential, as the potential difference tends to infinity. At the same time the incline to the t-axis and the aperture of the cone of \( t^{ - {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}\) -decay in the (t, x)-plane tend to zero.
Mathematics Subject Classification (2000). Primary 34B45; Secondary 47A70,35B40
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References
F. Ali Mehmeti, Spectral Theory and L ∞ -time Decay Estimates for Klein-Gordon Equations on Two Half Axes with Transmission: the Tunnel Effect. Math. Methods Appl. Sci. 17 (1994), 697–752.
F. Ali Mehmeti, Transient Waves in Semi-Infinite Structures: the Tunnel Effect and the Sommerfeld Problem. Mathematical Research, vol. 91, Akademie Verlag, Berlin, 1996.
F. Ali Mehmeti, R. Haller-Dintelmann, V. Régnier, Dispersive Waves with multiple tunnel effect on a star-shaped network; to appear in: “Proceedings of the Conference on Evolution Equations and Mathematical Models in the Applied Sciences (EEMMAS)”, Taranto 2009.
F. Ali Mehmeti, R. Haller-Dintelmann, V. Régnier, Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation. arXiv:1012.3068v1 [math.AP], preprint 2010.
F. Ali Mehmeti, V. Régnier, Delayed reflection of the energy flow at a potential step for dispersive wave packets. Math. Methods Appl. Sci. 27 (2004), 1145–1195.
J. von Below, J.A. Lubary, The eigenvalues of the Laplacian on locally finite networks. Results Math. 47 (2005), no. 3-4, 199–225.
S. Cardanobile and D. Mugnolo. Parabolic systems with coupled boundary conditions. J. Differential Equations 247 (2009), no. 4, 1229–1248.
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, Valenciennes, France, 2004.
J.M. Deutch, F.E. Low, Barrier Penetration and Superluminal Velocity. Annals of Physics 228 (1993), 184–202.
A. Enders, G. Nimtz, On superluminal barrier traversal. J. Phys. I France 2 (1992), 1693–1698.
A. Haibel, G. Nimtz, Universal relationship of time and frequency in photonic tunnelling. Ann. Physik (Leipzig) 10 (2001), 707–712.
L. Hörmander, The Analysis of Linear Partial Differential Operators I. Springer, 1984.
V. Kostrykin, R. Schrader, The inverse scattering problem for metric graphs and the travelling salesman problem. Preprint, 2006 (www.arXiv.org:math.AP/0603010).
O. Liess, Decay estimates for the solutions of the system of crystal optics. Asymptotic Analysis 4 (1991), 61–95.
B. Marshall, W. Strauss, S. Wainger, Lp -L q Estimates for the Klein-Gordon Equation. J. Math. Pures et Appl. 59 (1980), 417–440.
K. Mihalincic, Time decay estimates for the wave equation with transmission and boundary conditions. Dissertation. Technische Universität Darmstadt, Germany, 1998.
M. Pozar, Microwave Engineering. Addison-Wesley, New York, 1990.
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Mehmeti, F.A., Haller-Dintelmann, R., Régnier, V. (2012). The Influence of the Tunnel Effect on L ∞-time Decay. In: Arendt, W., Ball, J., Behrndt, J., Förster, KH., Mehrmann, V., Trunk, C. (eds) Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations. Operator Theory: Advances and Applications, vol 221. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0297-0_2
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