Large time behaviour of dispersive waves hitting a potential step above the threshold of tunnel effect
Prof. Dr. Felix Ali Mehmeti (Université de Valenciennes, Frankreich)

We consider Klein-Gordon Equations with piecewise constant coefficients on a star-shaped network. Using the spectral theory of the associated spatial operator developed in [A] and a version of the stationary phase method [D], we calculate the leading term of an asymptotic expansion for large times including an error estimate for solutions in energy bands [B]. The special case of two branches can be interpreted in physical terms for example as a model for a massive particle without spin (a pion) in a one dimensional world hitting a potential step at zero. We estimate the probability of the occurrence of the outgoing particle (in given energy bands) within or outside cones given by group lines in space time, i.e. the probability of local or nonlocal behaviour. Assuming that the particle energy band is above the threshold of tunnel effect, it turns out that

<ul> <li> the outgoing velocity for the locally behaving part of the particle is lower than the velocity expected from classical mechanics [B]. This effect is increased for high potential steps; <li> the ratio of the probabilities of local and nonlocal behaviour is mainly independent of the height of the potential step [C]; <li> the probability of transmission tends to zero when the potential step grows to infinity, while the energy excess with respect the threshold of tunnel effect is kept constant.
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The dependence of all results on the frequency (energy) band of the particle is given explicitly.

<p>There might be applications also in microwave physics and optics.

<p>References:

<p>[A] 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, J. Evol. Equ. 12 (2012), 513 - 545

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[B] F. Ali Mehmeti, R. Haller-Dintelmann, V. Régnier:
The influence of the tunnel effect on L-infinity time decay. W. Arendt, J. A. Ball, J. Behrndt, K.-H. Förster, V. Mehrmann, C. Trunk (eds): Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations: IWOTA10; Springer, Basel; Operator Theory: Advances and Applications, 221 (2012), 11 - 24.

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[C] F. Ali Mehmeti, R. Haller-Dintelmann, V. Régnier:
Energy flow above the threshold of tunnel effect. A. Almeida, L. Castro, F.-O. Speck (eds.): Advances in Harmonic Analysis and Operator Theory: the Stefan Samko Anniversary Volume; Springer, Basel; Operator Theory: Advances and Applications, 229 (2013), 65 -76.

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[D] L. Hörmander:
The Analysis of Linear Partial Differential Operators I. Springer, 1984