Time Delay for the Dirac Equation

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.