My goal is to express the result of the following integral in closed-form (using for example traces of the matrices):
$$ \int_{-\infty}^{\infty} b^T G(x)^T H G(x) b dx $$ where $$ b \in \mathbb{R}^{m \times 1} $$ $$ H \in \mathbb{R}^{m \times m} $$ $$ G = (xI - H)^{-1} $$
You may decompose the non-Hermitian operator $H$:
$$ H = \sum_m h_m \vert m \rangle \langle \tilde{m} \vert $$
and express the result in terms of the pseudoeigenvalues.