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.



Scattering of relativistic particle by long-range potential
It seems that long-range scattering for the operator $\sqrt{-\Delta+M^2}$ has not been dealt with in the literature. On the technical side, I would start with Chapter XXX of Hörmander [*The Analysis of Linear Partial Differential Operators IV*](http://www.springer.com/de/book/9783642001178), where long-range scattering for general $m$th-order elliptic differential operators is discussed.


There are a few papers on long-range scattering for Dirac operators, though. For instance, existence and completeness of the wave operators is treated by [Gâtel and Yafaev (2001)](http://www.sciencedirect.com/science/article/pii/S0022123601937425) which was improved by [Daudé (2005)](https://eudml.org/doc/116167) using time-dependent observables as introduced by Dereziński and Gérard [*Scattering Theory of Classical and Quantum $N$-Particle Systems*](http://www.springer.com/gp/book/9783540620662).


