Let $H_{eff}$ be a $n$ dimensional matrix defined by the eigenvalue spectrum $\Lambda$: $$\Lambda(H)_n=\Lambda(H_n+H_{eff}),$$ Where $H$ is a infinite dimensional matrix, $\Lambda(H)_n$ are its lowest $n$ eigenvalues and $H_n$ is the $n$ dimensional sub-matrix of $H$ with the $n$ smallest diagonal elements ($H$ is almost diagonal).
$H$ is hermitian, so $\Lambda\in\mathbb{R}$. On the r.h.s., $H_n$ is also Hermitian, nevertheless $H_{eff}$ may not be Hermitian (A non-Hermitian Matrix can still produce a real eigenvalue spectrum).
My problem is the following: $H_n$ is well defined and I did have a discription of $H_{eff}$. But $H_{eff}=R+C$ is non-Hermitian as $R^\dagger=R$, $C^\dagger=-C$. Calculating the eigenvalue spectrum of Hermitian matrices is much faster.
Is there some kind of expantion in which I approach the real spectrum, but in each step, I only have to calculate the eigenvalues of hermitan Matrices?
(It might be imortant, that $H_{eff}$ is given in a perturbative expansion $H_{eff}^{(i)}=R^{(i)}+C^{(i)}$, such that $\delta\lambda_j<(\lambda_j-\lambda_n)^{-i}$. And $C<<R$, e.g at first order I found putting C=0 produces errors of order 2. However, I would prefere to have these expansions indipendent)
Answers and ideas would be great, but references are most welcome as well.
