abstract

Felix Ali Mehmeti
Tunnel Effect and Sommerfeld Problem:
Transient Waves in Semi-Infinite Structures
Summary
We study transient waves in media with semi-infinite geometry in two significant cases.
The models are linear and describe 1. the collision of a relativistic massive particle with
a semi-infinite potential step in one space dimension and 2. the diffraction of waves at a
semi-infinite crack in a two-dimensional, homogeneous, vibrating medium (the ‘Sommerfeld Problem’).
We derive for problem 1 an estimate for the L∞ −time decay rate of the solution which
is reduced as compared with the collision free case. To this end we expand the solution
in generalized eigenfunctions and use the method of stationary phase. The result is
connected with the phenomenon of violation of causality observed for tunnelling particles
in recent experiments.
For problem 2 we obtain for real wave numbers a representation of stationary solutions
in terms of generalized eigenfunctions and a limiting absorption principle. We use formulae
of E. Meister and F.-O. Speck for the absorption case, uncertainty principles for Laplace
integrals, features of spectral theory and the method of stationary phase. Our result solves
an open problem and is an important step towards a solution formula for the transient
problem.
Short description
We investigate the L∞ −time decay of solutions of the Klein-Gordon equation in one
space dimension with a semi-infinite potential step and derive a limiting absorption principle for the wave equation in two space dimensions with zero Dirichlet conditions imposed
on a half axis. These are model cases for transient waves in media with semi-infinite
geometry involving interfaces and crack-tips.
Kurztext
Wir untersuchen das zeitasymptotische Verhalten eines Modells für die Kollision eines
relativistischen, massiven Teilchens mit einer halbunendlichen Potentialschwelle und gewinnen ein Grenzabsorptionsprinzip für die Beugung von Wellen an einem halbunendlichen
Riß in einem unendlich ausgedehnten, homogenen, vibrierenden Medium. Dies sind Modellfälle für transiente Wellen in Medien mit halbunendlicher Geometrie. Hier liegt die
Strategie zugrunde, die Untersuchung komplizierterer Medien durch Lokalisierung in der
Nähe von Grenzflächen oder Rißspitzen auf das Studium einfacher halbunendlicher Konfigurationen zurückzuführen.
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From the Contents
1. Spectral theory and L∞ -time decay for the Klein-Gordon equation with potential
step: the Tunnel Effect
• Expansion in generalized eigenfunctions
• L∞ -time decay estimates
• Physical interpretations
2. A limiting absorption principle for the Sommerfeld Problem in the plane
• A formula for the resolvent extendible for real wave numbers
• Generalized eigenfunctions as Laplace integrals
• An uncertainty principle for some Laplace integrals
• A limiting absorption principle
• Convergence speed of the limiting absorption process
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