How to solve a Klein-Gordon/Helmholtz scattering problem with a static vector potential and obtain the near-field scattering wavefunction?

I am studying the scattering of a charged scalar particle in a prescribed static vector-potential background $\mathbf A(\mathbf r)$. After separating the time dependence, the stationary wavefunction satisfies a Klein--Gordon/Helmholtz-type equation
\begin{equation}
\left[(\nabla-iq\mathbf A)^2+k^2\right]\psi(\mathbf r)=0 ,
\end{equation}
where $q$ is the coupling to the external vector potential and $k$ is the incoming momentum.

The vector potential $\mathbf A(\mathbf r)$ is a given static spatial function. It is significant near a region of size $R$, and decays away from that region.

My goal is not primarily to compute the far-field scattering amplitude $f(\theta,\phi)$, but rather to obtain the near-field scattering wavefunction itself, especially
\begin{equation}
\psi(r\simeq R,\theta,\phi)
\end{equation}
and the angular average
\begin{equation}
\left\langle |\psi(R,\theta,\phi)|^2 \right\rangle_{\Omega}.
\end{equation}
In other words, I want to know whether the external vector potential suppresses, enhances, shadows, or focuses the wavefunction near the interaction region.

Equation

Expanding the operator gives
\begin{equation}
\nabla^2\psi

• 2iq\mathbf A\cdot\nabla\psi
• iq(\nabla\cdot\mathbf A)\psi
• q^2A^2\psi
• k^2\psi=0 .

\end{equation}

In the Coulomb gauge, $\nabla\cdot\mathbf A=0$, this becomes
\begin{equation}
\nabla^2\psi

• 2iq\mathbf A\cdot\nabla\psi
• q^2A^2\psi
• k^2\psi=0 .

\end{equation}

The incident wave may be taken as
\begin{equation}
\psi_{\rm inc}=e^{i\mathbf k\cdot\mathbf r}.
\end{equation}

I want to solve for the total wavefunction
\begin{equation}
\psi=\psi_{\rm inc}+\psi_{\rm sc},
\end{equation}
where the scattered wave satisfies an outgoing radiation condition.

Parameter regime

The regime I am interested in satisfies
\begin{equation}
kR\gg1 ,
\end{equation}
for example
\begin{equation}
kR\sim 10^2-10^3 .
\end{equation}

Therefore this is not a low-energy scattering problem. A standard partial-wave expansion may require a very large $\ell_{\max}$, and for strong coupling even more angular modes may be needed.

On the other hand, solving directly for $\psi$ using 3D FEM/FDM may also be difficult because $\psi$ is highly oscillatory.

Questions

1. What is the most suitable numerical method for this problem?

Should one use partial-wave/R-matrix methods, finite element methods, boundary element or FEM--BEM hybrid methods, eikonal/WKB/ray tracing methods, or a phase-extracted FEM method?

2. Should the fast incident phase be factored out?

For example, write
\begin{equation}
\psi(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u(\mathbf r),
\end{equation}
and solve only for the slowly varying envelope $u(\mathbf r)$.

Then, defining
\begin{equation}
\mathbf Q(\mathbf r)=\mathbf k-q\mathbf A(\mathbf r),
\end{equation}
one obtains a weak form schematically of the form
\begin{equation}
\int_\Omega
(\nabla v+i\mathbf Qv)^*
\cdot
(\nabla u+i\mathbf Qu)\,dV
k^2\int_\Omega v^*u\,dV
\text{boundary terms}.
\end{equation}

Is this phase-extracted FEM formulation more stable than solving directly for $\psi$?

If $q\mathbf A$ is strong, should one instead use an eikonal phase
\begin{equation}
\psi=e^{iS_0(\mathbf r)}u,
\end{equation}
where $S_0$ solves the Hamilton--Jacobi equation
\begin{equation}
(\nabla S_0-q\mathbf A)^2=k^2?
\end{equation}

3. How should the scattering boundary condition be implemented?

If the computational domain is truncated at $r=R_{\rm out}$, what boundary condition should be used?

A simple option is a Sommerfeld/Robin condition,
\begin{equation}
\mathbf n\cdot(\nabla-iq\mathbf A)\psi-ik\psi
\mathbf n\cdot(\nabla-iq\mathbf A)\psi_{\rm inc}
• ik\psi_{\rm inc}.
\end{equation}

But in the regime $kR\gg1$, is such a local boundary condition sufficient? Should one instead use PML, a spherical DtN map, or FEM--BEM coupling?

4. If I only care about the near-field wavefunction, is there a more direct approach?

Standard scattering theory often focuses on the far-field form
\begin{equation}
\psi
\sim
e^{i\mathbf k\cdot\mathbf r}
+
f(\theta,\phi)\frac{e^{ikr}}{r}.
\end{equation}

However, I need
\begin{equation}
\psi(r\simeq R,\theta,\phi).
\end{equation}

In this case, is it still necessary to construct the full far-field scattering solution accurately? Or can one use a finite-radius outgoing boundary condition and directly solve for the near field?

5. How can one validate the result in the strong-coupling regime?

For large $qAR$, the wavefunction may exhibit focusing, shadowing, caustics, local enhancement, local suppression, or strong angular structure.

What convergence and consistency checks should be performed? For example:

1. $R_{\rm out}$ convergence;

2. $h$-refinement;

3. $p$-refinement;

4. PML thickness convergence;

5. comparison with ray tracing;

6. the $q\to0$ free-wave limit.

Desired outcome

I am looking for a robust numerical strategy for computing the near-field scattering state of
\begin{equation}
\left[(\nabla-iq\mathbf A)^2+k^2\right]\psi=0,
\end{equation}
especially in the high-frequency regime
\begin{equation}
kR\gg1
\end{equation}
with possibly strong vector-potential coupling.

In particular, I would like advice on:

1. whether partial-wave methods should be avoided in this regime;

2. whether phase-extracted FEM is the right formulation;

3. how to impose the outgoing boundary condition;

4. whether there are existing Helmholtz/FEM/BEM codes or techniques suitable for this;

5. how to verify the reliability of the computed near-field $|\psi|^2$.