# The problem of embedded eigenvalues for the Dirac equation in the Schwarzschild black hole metric

## TL;DR

This paper proves that there are no bound fermionic states, including embedded eigenvalues, around a Schwarzschild black hole, correcting previous claims based on oversimplified models.

## Contribution

The study rigorously demonstrates the absence of embedded eigenvalues for the Dirac equation in Schwarzschild spacetime, clarifying misconceptions from earlier approximations.

## Key findings

- No embedded eigenvalues exist for energies below the particle's mass.
- Previous claims of bound states are due to oversimplified calculations.
- Fermionic bound states are absent regardless of particle mass.

## Abstract

We use the Dirac equation in a fixed black hole background and different independent techniques to demonstrate the absence of fermionic bound states around a Schwarzschild black hole. In particular, we show that no embedded eigenvalues exist which have been claimed for the case when the energy is less than the particle's mass. We explicitly prove that the claims regarding the embedded eigenvalues can be traced back to an oversimplified approximation in the calculation. We conclude that no bound states exist regardless the value of the mass.