# Fermionic non-zero modes to standard $SU(2)$ 't Hooft instantons?

The fermionic zero modes to the "standard" $$SU(2)$$ 't Hooft instantons are known as follows (see page 42 in this article):

For a 't Hooft instanton (self-dual field strength tensor) in the regular gauge, centered at the origin, the massless Dirac equation admits a unique right-handed normalizable fundamental-representation Weyl spinor solution (up to an overall constant):

$$\psi_{\alpha,a}(x)=\begin{pmatrix}0 \\ \lambda_{\alpha,a}(x) \end{pmatrix}\,\,\, \textrm{(Right-handed)}$$

$$\lambda_{\alpha,a}(x)=\frac{\epsilon_{\alpha a}}{(x^2+\rho^2)^{3/2}}$$

where $$\alpha=1,2$$ is the spinor index, $$a=1,2$$ is the color/gauge index, and $$\epsilon_{\alpha a}$$ is the standard Levi-Civita tensor which surprisingly mixes spinor and color indices. By definition, this solution satisfies the equation:

$$(\sigma^{\mu})^{\alpha \alpha'}(\partial_\mu \lambda_{\alpha', a}-iA^{m}_{\mu}(T_m)_{ab}\lambda_{\alpha', b})=0$$

where $$A^m_{\mu}$$ is the 't Hooft instanton gauge field, $$T_m$$ are the 3 $$SU(2)$$ infinitesimal generators (here in the fundamental representation, so they're just Pauli matrices), and $$\sigma^\mu=(\sigma^i,i)$$.

In the singular gauge, the corresponding solution is:

$$\lambda_{\alpha,a}(x)=\frac{1}{x (x^2+\rho^2)^{3/2}}x_{\mu}(\sigma^{\dagger \mu})_{\alpha,a}$$

[My Question] What about non-zero modes? In other words, what is the spectrum of the Dirac operator in the background of an $$SU(2)$$ self-dual instanton? ($$\gamma^{\mu}D_{\mu}\psi=\lambda \psi$$)

Do the corresponding fermionic non-zero modes have analytic solutions? What about orthogonality relations between these eigenfunctions?

I'm having a hard time finding answers to this online...

I am not aware of any attempts to write down the non-zero mode solutions. However, we are (typically) not really interested in individual non-zero mode solutions, but in the non-zero mode propagator $$S(x,y)=\sum_{\lambda\neq 0} \frac{\psi_\lambda(x)\psi_\lambda^\dagger(y)}{\lambda}$$ and this object has been determined, see here.