Let
$\mathcal{H}=L^2(\mathbb{R}^3)$,
$H_0=\sqrt{-\Delta+M^2}$, ($M$ is a positive constant, $\Delta$ is the laplacian)
and
$H=H_0+V(\vec{x})$
(where $V(\vec{x})$ is the operator of multiplication by a bounded continuous function $V(\vec{x})$; $H$ is a perturbation of $H_0$ by a bounded operator).
Consider the operators $$\exp(it H)\exp(-itH_0)\Psi, $$ where $\Psi\in\mathcal{H}$.
If $V$ is square-integrable then it is relatively easy to show (using the Cook's method) that the limit $$\Omega_\pm=\lim_{t\to\pm\infty}\exp(it H)\exp(-it H_0)\Psi$$ exists.
If $V(\vec{x})$ is a long-range potental, i.e. it vanishes in infinity not faster than the Coulomb potential $(1/|\vec{x}|)$ than the above limit does not exist. The above expression has to be modified.
Scattering of a non-relativistic particle by a long-range potential: If $H_0=-\Delta$ than $$\Omega_\pm=\lim_{t\to\pm\infty}\exp(it H)U_{as}(t)\Psi$$ exists, where $U_{as}$ is the the asymptotic evolution operator (describing 'distorted free' propagation of a particle in a potential $V$). This generalization was first proposed by Dollard in case of the potential $V(\vec{x})=\frac{const}{|\vec{x}|}$. Nowadays the theory of scattering of a non-relativistic particle by a long-range potential is well-developed.
I'm interested in a similar results in the relativistic case, i.e. $H_0=\sqrt{-\Delta+M^2}$ and $V$ is a long-range potential (vanishing in infinity like Coulomb potential). Is there a 'modified free' evolution operator $U_{as}(t)$ such that $$\Omega_\pm=\lim_{t\to\pm\infty}\exp(it H)U_{as}(t)\Psi$$ exists.
I will be grateful for any references and comments.
