Levinson's Theorem for Dirac Equation

TL;DR

This paper proves a stronger version of Levinson's theorem for the Dirac equation, relating separate positive and negative energy phase shifts to bound states of an associated Schrödinger equation, revealing new insights into quantum scattering.

Contribution

It introduces a novel formulation of Levinson's theorem for Dirac equations, connecting phase shifts to bound states of a related Schrödinger problem, enhancing understanding of quantum scattering.

Findings

Separate phase shifts for positive and negative energies relate to specific bound states.

The relation involves a Schrödinger equation that matches the Dirac equation at zero momentum.

The theorem provides a more detailed link between phase shifts and bound states than previous versions.

Abstract

Levinson's theorem for the Dirac equation is known in the form of a sum of positive and negative energy phase shifts at zero momentum related to the total number of bound states. In this letter we prove a stronger version of Levinson's theorem valid for positive and negative energy phase shifts separately. The surprising result is, that in general the phase shifts for each sign of the energy do not give the number of bound states with the same sign of the energy (in units of $\pi$), but instead, are related to the number of bound states of a certain Schr\"odinger equation, which coincides with the Dirac equation at zero momentum.