$H_0=\sqrt{-\Delta+M^2}$, ($M$ is a positive constant, $\Delta$ is the laplacian)



(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.

  • 1
    $\begingroup$ Why doesn't Hörmander's method apply to $H_0$ ? $\endgroup$
    – jjcale
    Jan 28, 2017 at 10:21
  • $\begingroup$ Could you elaborate on this? Does this method apply to $H_0=(-\Delta+m^2)^{1/2}$ $\endgroup$
    – user72829
    Jan 28, 2017 at 12:01

1 Answer 1


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, 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) which was improved by Daudé (2005) using time-dependent observables as introduced by Dereziński and Gérard Scattering Theory of Classical and Quantum $N$-Particle Systems.


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