Question

Let $\Omega\subset\mathbb{R}^n$ be bounded with smooth boundary, $d_i>0, f_{ij}\in L_{\infty}(\Omega)$ Consider operator $A:(L_2(\Omega))^2\to(L_2(\Omega))^2$ with domain $D(A)=(H^2_N(\Omega))^2$ ('N' stands for Neumann bcd)

\begin{align} A^1(u,v)=d_1\Delta u+f_{11}(x)u+f_{12}(x)v \quad A^2(u,v)=d_2\Delta v+f_{21}(x)u+f_{22}(x)v \end{align}

1. Is there any elegant way to express the spectrum of $A$ in terms of $d_i, f_{ij}$? From the compactness of the resolvent I know that spectrum is pure point. I do not assume that $f_{12}=f_{21}$, so in general the spectrum will not be real.

2. The same for the case $d_1>0, d_2=0$ (then the spectrum does not have to be even pure point).

Thanks for any comments.