Time delay for the Dirac equation

TL;DR

This paper introduces a novel method to analyze the asymptotic behavior of expectation values related to the Dirac equation, enabling the calculation of the time delay operator and its relation to spectral shift functions.

Contribution

A new approach for computing the asymptotics of expectation values in the Dirac equation, leading to explicit expressions for the time delay operator and its spectral relations.

Findings

Derived the asymptotics of expectation values for the free Dirac operator.

Established the relation between the time delay operator and the Eisenbud-Wigner operator.

Connected the averaged time delay with the spectral shift function.

Abstract

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator $\int\limits_{0} ^{\infty}e^{iH_{0}t}\zeta\left( \frac{\left\vert x\right\vert }{R}\right) e^{-iH_{0}t}dt,$ as $R\rightarrow\infty,$ is presented. Here $H_{0}$ is the free Dirac operator and $\zeta\left( t\right) $ is such that $\zeta\left( t\right) =1$ for $0\leq t\leq1$ and $\zeta\left( t\right) =0$ for $t>1.$ This approach allows us to obtain the time delay operator $\delta \mathcal{T}\left( f\right) $ for initial states $f$ in $\mathcal{H} _{2}^{3/2+\varepsilon}\left( \mathbb{R}^{3};\mathbb{C}^{4}\right) ,$ $\varepsilon>0,$ the Sobolev space of order $3/2+\varepsilon$ and weight $2.$ The relation between the time delay operator $\delta\mathcal{T}\left( f\right) $ and the Eisenbud-Wigner time delay operator is given. Also, the relation between the averaged time delay and the spectral shift function is presented.