Energy flow above the threshold of tunnel effect

TL;DR

This paper analyzes the energy propagation of solutions to the Klein-Gordon equation on a two-branch domain with different potentials, focusing on behavior above the tunnel effect threshold and the impact of increasing potential differences.

Contribution

It applies stationary phase methods to study energy flow above the tunnel effect threshold and investigates the asymptotic behavior as potential differences grow large.

Findings

A fixed portion of energy propagates between group lines above the tunnel threshold.

Total transmitted energy decreases inversely with the square root of the potential difference.

The cone of group lines narrows and inclines towards the t-axis as potential difference increases.

Abstract

We consider the Klein-Gordon equation on two half-axes connected at their origins. We add a potential that is constant but different on each branch. In a previous paper, we studied the L-infinity-time decay via H\"ormander's version of the stationary phase method. Here we apply these results to show that for initial conditions in an energy band above the threshold of the tunnel effect a fixed portion of the energy propagates between group lines. Further we consider the situation that the potential difference tends to infinity while the energy band of the initial condition is shifted upwards such that the particle stays above the threshold of tunnel effect. We show that the total transmitted energy as well as the portion between the group lines tend to zero like the inverse of the square root of the higher potential in the corresponding branch if this potential tends to infinity. At the same time the cone formed by the group lines inclines to the t-axis while its aperture tends to zero.