MULTIPLE TUNNEL EFFECT FOR DISPERSIVE WAVES ON A STAR SHAPED NETWORK: AN EXPLICIT FORMULA FOR THE

MULTIPLE TUNNEL EFFECT FOR DISPERSIVE WAVES ON A STAR-SHAPED NETWORK: AN EXPLICIT FORMULA FOR THE SPECTRAL REPRESENTATION F. ALI MEHMETI, R. HALLER-DINTELMANN, AND V. REGNIER Abstract. 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. Further we prove the surjectivity of the associated transformation, thus showing that it is in fact a spectral representation. The characteristics of the problem are marked by the non-manifold character of the star- shaped domain. Therefore the approach via the Sturm-Liouville theory for systems is not well-suited. The considerable effort to construct explicit formulas involving the tunnel effect generalized eigenfunctions is justified for example by the perspective to study the influence of tunnel effect on the L∞-time decay. 1. Introduction This paper is motivated by the attempt to study the local behavior of waves near a node in a network of one-dimensional media having different dispersion properties. This leads to the study of a star-shaped network with semi-infinite branches.

• defined function


• spectral theory


• problem can


• symmetrization problem


• operator defined


• tunnel effect


• transport operator


• klein-gordon equations


• branche


• given spectral