Dirac operators on compact manifolds seem to have been studied well, such as in this book and also this one.

However, I cannot easily find comprehensive treatment of Dirac operators coupled to gauge potentials as appearing in Fujikawa method. There is a paper by Atiyah dealing with this topic, but it is extremely terse and not easy to read.

Moreover, noncompact manifolds like $\mathbb{R}^n$ are not usually chosen as the underlying space for such operators.

To be more specific, my question is as such:

On the $\mathbb{R}^4$ as the Euclidean space, is there a choice of the vector potential $A$ such that the Dirac operator $D:= \gamma^j \partial_j + i A_{\alpha j}t^\alpha \gamma^j$ is a self-adjoint operator in a suitable Hilbert space having some orthonormal eigenbasis with Schwartz function components?

Here, $\gamma^j$ are the (Euclidean) gamma matrices and $t^\alpha$ are normalized generators of some Lie algebra $\mathfrak{g}$, typically $\mathfrak{su}(2)$ or $\mathfrak{su}(3)$. $j=1,2,3,4$ are the spacetime indices and $\alpha$ are Lie algebra indices. The summation convention is assumed here.

This question seems closely related to characterization of Schwartz functions on the sphere and asymptotic behavior at infinity associated with nontrivial topology.

Could anyone please help me with this topic? If the information I have provided above does not look sufficient, please comment. I appreciate any feedback.