The Influence of the Tunnel Effect on L-infinity-time decay

TL;DR

This paper investigates how the tunnel effect influences the decay rate of solutions to the Klein-Gordon equation on a star-shaped network, revealing that potential differences can diminish the leading decay coefficient over time.

Contribution

It provides a spectral theoretic analysis of L-infinity decay for the Klein-Gordon equation on networks, highlighting the impact of the tunnel effect on decay coefficients.

Findings

The leading decay coefficient tends to zero as potential difference increases.

The decay cone in the (t,x)-plane narrows with increasing potential difference.

The decay rate follows a t^{-1/2} asymptotic behavior influenced by the tunnel effect.

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. Exploiting a spectral theoretic solution formula from a previous paper, we study the L-infinity-time decay via H\"ormander's version of the stationary phase method. We analyze the coefficient c of the leading term c t^{-1/2} of the asymptotic expansion of the solution with respect to time. For 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^{-1/2}-decay in the (t,x)-plane tend to zero.