Scattering state and bound state of scalar field in Schwarzschild spacetime: Exact solution

TL;DR

This paper provides exact solutions for scalar fields in Schwarzschild spacetime, including scattering and bound states, along with explicit eigenvalues and phase shifts, and analyzes horizon discontinuities.

Contribution

It offers the first exact solutions for scalar field states in Schwarzschild spacetime and explicit formulas for eigenvalues and phase shifts, enhancing understanding of horizon behavior.

Findings

Exact wave functions for scattering and bound states

Explicit approximate eigenvalues and phase shifts

Direct calculation of horizon discontinuity

Abstract

The main aim of this paper is twofold. (1) Exact solutions of a scalar field in the Schwarzschild spacetime are presented. The exact wave functions of scattering states and bound-states are presented. Besides the exact solution, we also provide explicit approximate expressions for bound-state eigenvalues and scattering phase shifts. (2) By virtue of the exact solutions, we give a direct calculation for the discontinuous jump on the horizon for massive scalar fields, while in literature such a jump is obtained from an asymptotic solution by an analytic extension treatment.