to the action related to the derivative of $H_3$ in Eqs.(\[sing\_gauge\],\[surf\_gauge\]).

A combination of (\[cont\_cond\_sc\]) and (\[conditions\]) obviously leads to the relation $$\begin{aligned}
 \label{relations}
\phi_{k}^{(\pm)}(x_1,x_2)=\phi_{k}^{(\pm)}(-x_1,-x_2),\end{aligned}$$ where $(\pm)$ denotes the duality of the background field for a given $x_1$. This identity allows one to show that the eigenfunctions $$\Phi_{kl}(x) = \left\{ 
\begin{array}{l}
\Phi^{(+)}_{kl}(x), \, \, x_1\in L_+\\
\Phi^{(-)}_{kl}(x), \, \, x_1\in L_-
\end{array} 
\right., \, k,l=0,1\dots$$ form a complete orthogonal set in the space of square integrable functions which are even with respect to simultaneous reflection $x_1\to -x_1$ and $x_2\to-x_2$.

The eigenfunctions are of the bound state type with the purely discrete spectrum. Field fluctuations of this type can be seen as confined. It should be noted that the eigenvalues coincide with those for the purely homogeneous (anti-)self-dual Abelian field. In this sense, the domain wall defect does not destroy dynamical confinement of color charged fields. The eigenfunctions are restricted by the correlated evenness condition (\[cont\_cond\_sc\]), while in the case of the homogeneous field the properties of the amplitude $f(p_1)$ and the polynomial $H_k$ are mutually independent.

Color charged quasiparticles on the wall
----------------------------------------

Let us now consider the eigenvalue problem on the domain wall, i.e. for the region $x_1=0$. On 