# Scalar field in Reissner-Nordstr\"om spacetime: Bound state and scattering state

## TL;DR

This paper investigates the behavior of a massive scalar field in Reissner-Nordström spacetime, deriving wave functions, eigenvalues, and phase shifts for bound and scattering states, with a focus on the tortoise coordinate.

## Contribution

It provides explicit solutions for scalar field bound and scattering states in Reissner-Nordström spacetime, including phase shifts using the integral equation method.

## Key findings

- Derived bound-state wave functions and eigenvalues.
- Obtained explicit scattering phase shift expressions.
- Introduced the tortoise coordinate for analysis.

## Abstract

In this paper, we solve the massive scalar field in the Reissner-Nordstr\"om spacetime. The scalar field in the Reissner-Nordstr\"om spacetime has both bound states and scattering states. For bound states, we solve the bound-state wave function and the eigenvalue spectrum. For scattering states, we solve the scattering wave function and give an explicit expression for scattering phase shift by the integral equation method. Especially, we introduce the tortoise coordinate for the Reissner-Nordstr\"om spacetime.