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 $\underset{0}{\overset{\infty }{\int }}{e}^{i{H}_{0}t}\zeta \left(\frac{|x|}{R}\right)e^{-i{H}_{0}t}dt,$ as $R\to \infty ,$ is presented. Here $H_0$ is the free Dirac operator and $\zeta \left(t\right)$ is such that $\zeta \left(t\right)=1$ for $0\le t\le 1$ 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 ${\mathcal{H}}_2^{3/2+\epsilon }\left({\mathbb{R}}^3;{\mathbb{C}}^4\right),$ $\epsilon >0,$ the Sobolev space of order $3/2+\epsilon$ 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.

References in corpus (1)

• High-energy and smoothness asymptotic expansion of the scattering amplitude for the Dirac equation and applications

Cited by in corpus (1)

• Quantum time delay for unitary operators: general theory