Scattering by a long-range potential

Abstract

The phenomenon of wave tails has attracted much attention over the years from both physicists and mathematicians. However, our understanding of this fascinating phenomenon is not complete yet. In particular, most former studies of the tail phenomenon have focused on scattering potentials which approach zero asymptotically ($x\to \infty$) faster than $x^{-2}$. It is well-known that for these (rapidly decaying) scattering potentials the late-time tails are determined by the first Born approximation and are therefore {\it linear} in the amplitudes of the scattering potentials (there are, however, some exceptional cases in which the first Born approximation vanishes and one has to consider higher orders of the scattering problem). In the present study we analyze in detail the late-time dynamics of the Klein-Gordon wave equation with a ({\it slowly} decaying) Coulomb-like scattering potential: $V(x\to \infty )=\alpha /x$. In particular, we write down an explicit solution (that is, an exact analytic solution which is not based on the first Born approximation) for this scattering problem. It is found that the asymptotic ($t\to \infty$) late-time behavior of the fields depends {\it non}-linearly on the amplitude $\alpha$ of the scattering potential. This non-linear dependence on the amplitude of the scattering potential reflects the fact that the late-time dynamics associated with this slowly decaying scattering potential is dominated by {\it multiple} scattering from asymptotically far regions.

5 pages

References in corpus (8)

• Quasinormal modes of black holes: from astrophysics to string theory
• Stability and thermodynamics of brane black holes
• High accuracy simulations of Kerr tails: coordinate dependence and higher multipoles
• Late-time evolution of charged massive Dirac fields in the Kerr-Newman background
• Decay of Dirac Massive Hair in the Background of Spherical Black Hole
• Quasinormal modes of black holes absorbing dark energy
• How pure is the tail of gravitational collapse ?
• Anomalously small wave tails in higher dimensions

Cited by in corpus (8)

• Kerr-Newman black holes with stationary charged scalar clouds
• Exact solution of inverse-square-root potential $V\left(r\right)=-\frac{\alpha }{\sqrt{r}}$
• Scattering state and bound state of scalar field in Schwarzschild spacetime: Exact solution
• Scattering theory without large-distance asymptotics in arbitrary dimensions
• Scalar scattering in Schwarzschild spacetime: Integral equation method
• Long-range potential scattering: Converting long-range potential to short-range potential by tortoise coordinate
• Scalar field in Reissner-Nordström spacetime: Bound state and scattering state
• Scattering theory without large-distance asymptotics: scattering boundary condition