SPECTRAL THEORY AND L-INFINITY-TIME DECAY-ESTIMATES FOR KLEIN-GORDON EQUATIONS ON 2 HALF AXES WITH TRANSMISSION - THE TUNNEL EFFECT

Consider two copies N1, N2 of the interval [0, infinity). Consider Klein-Gordon equations with (different) constant coefficients on R x N(j)(= time x space). Assume the coincidence of the values of the solution at the boundary points of the N(j) for all times and a transmission condition relating its first (one-sided) space derivatives at these points. Under a symmetry condition, we extend the spatial part of the equation and the transmission conditions to a self-adjoint operator (by Friedrichs extension) and reformulate our problem in terms of an abstract wave equation in a suitable Hilbert space. We derive an expansion of the solution in generalized eigenfunctions of this self-adjoint extension and show, that the L(infinity)-norms (in space) of the solution and its first k space derivatives at the time t decay for t --> infinity at least as const. t-1/4, if the initial conditions satisfy a compatibility condition of order k derived in this paper. The loss of decay rate in comparison with the full line case (const. t - 1/2, d [28]) is caused by the tunnel effect. Further we show that an abstract wave equation in a Hilbert space with a Friedrichs extension as spatial part can always be derived from a stationarity principle for an associated action-type functional. This yields a physical legitimation of our model by the principle of stationary action and moreover a criterion for the physical interpretability of all models created by the linear interaction concept [4, 6, 8, 10), in particular for the coupling of media of different dimension (alternative to [13, 16] for similar models).