This might be considered as a continuation of my previous question Spectrum of a linear elliptic operator but is independent. I have another question on V. Gribov's paper "Quantization of non-Abelian gauge theories". Nuclear Physics B. 139: 1–19.
Let $\frak{g}$ be a Lie algebra of a compact simple Lie group (e.g. $\frak{g}=su(2)$ is interesting enough). Let $$A_\mu\colon \mathbb{R}^4\to \frak{g}, \mu=1,2,3,4.$$ be smooth functions of fast decay such that $$ \partial_\mu A_\mu=0 \,\,\,\,\, (1) $$ (with summation convention over repeated indices).
Consider the following linear elliptic operator on maps $\alpha\colon \mathbb{R}^4\to \frak{g}$:
$$L_A(\alpha)=-\Delta\alpha +[A_\mu,\partial_\mu \alpha],$$ where $\Delta$ is the usual Laplacian acting componentwise, $[\cdot,\cdot]$ is the Lie bracket. Due to (1) the operator $L_A$ is self adjoint.
V. Gribov makes the following claim without any explanation (see section 3 of his paper though he is using a different notation):
There exist smooth $A_\mu$ with fast decay at infinity such that the discrete spectrum of $L_A$ contains 0.
I would like to have a proof of the above claim.
Remark. This claim is made on p. 5 of this paper https://reader.elsevier.com/reader/sd/pii/055032137890175X?token=E9E4528EF06235A698490920BA853B52363405F5D75E83E2B413C76CC74EE9A5977899AA8B8A09DEDF85780F6F703653
