I have a general non-self adjoint operator $L$, and a set of equations with some inhomogeneous boundary conditions,
$$ Lq= 0 \quad in \quad \Omega, $$
$$ q = q_0 \quad on \quad \partial\Omega. $$
I am studying the sensitivity of the solution with respect to the boundary conditions. I have been using Lagrangian approach, but I would really like to get some physical insight on how the boundary condition affects the state $q$.
I know that the Green's function is essentially an operator that provides such information. However, since this is a numerical study, my guess would be to present the Green's function in terms of basis functions: $$ G(x,x') = \sum \frac{q_n(x) q_n(x')}{\lambda_n}. $$
What happens if the original operator is non self-adjoint? In fact, it is $\textit{almost}$ self-adjoint, i.e. is self-adjoint up to a linear transformation of the state vector $q$.
Can I do some kind of eigenvalue decomposition of the original and the adjoint problems to present the Green's function as a sum of basis functions product?