Dirac operators on compact manifolds seem to have been studied well, such as in [this book](https://books.google.co.uk/books/about/Heat_Kernels_and_Dirac_Operators.html?id=_e2FjvLbO94C&redir_esc=y) and also [this one](https://bookstore.ams.org/gsm-25/).


However, I cannot easily find comprehensive treatment of Dirac operators coupled to gauge potentials as appearing in [Fujikawa method](https://en.wikipedia.org/wiki/Fujikawa_method). There is a [paper by Atiyah](https://www.jstor.org/stable/23378) 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](https://en.wikipedia.org/wiki/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.


This question seems closely related to [characterization of Schwartz functions on the sphere](https://math.stackexchange.com/questions/4874825/is-it-really-true-that-mathcals-mathbbrn-is-identified-with-smooth-fun) and [asymptotic behavior at infinity associated with nontrivial topology](https://physics.stackexchange.com/questions/815752/topological-behavior-or-asymptotics-at-infinity-of-gauge-fields-assumed-in-fuj).

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.