I am looking to understand the spectrum of the following operator: $L_{\epsilon}[u](x)=L_0[u](x)+i\epsilon x u(x)$ on $\mathbb R$. Here $L_0$ is a negative semi-definite self-adjoint operator that has a double zero eigenvalue, and all other eigenvalues are negative (& real).
Q1: Is this a well-studied problem ? Can I just use regular perturbation theory on this type of problem ?
Let $\lambda^0_i,v^0_i(x)$ be the eigenvalues and eigenfunctions of $L_0$. Then can I expand $\lambda^{\epsilon}_i=\lambda^0_i+\epsilon l_{i,1}+\epsilon^2 l_{i,2}+....$, and $v^{\epsilon}(x)_i=v^0_i(x)+\epsilon v_{i,1}(x)+\epsilon^2 v_{i,2}(x)+....$, and proceed with regular perturbation theory ?
When I do as above, I find that the perturbation theory fails to accurately predict the imaginary part of the first couple of eigenvalues that are perturbations of the double zero eigenvalues.
Q2: Can we prove the above operator has a discrete spectrum ? In other words, can we prove that $R_{\lambda}=(\lambda-L_{\epsilon})^{-1}$ is a compact operator ?