How to find the eigenvalues of $O$ in the following integral equation?

$$O=expleft(x^TAx +x’^T B x’+x^TCx’+x’^TDxright)$$

Where $x^T=(x_0,x_1,..,x_{n-1})$, $x’^T=(x’_0,x’_1,..,x’_{n-1})$ and $A,B,C,D$ are complex square $ntimes n$ matrices. The column matrices x and x’ consists of real entries. Also, $A^dagger=B$ and $C&D$ are Hermitian. I need to know a proper eigenfunction to solve for $lambda_n$the following equation:
$$int_{-infty}^{+infty} O f(x’)_nprod_{i=0}^{n-1} dx’_i=lambda_n f(x)_n$$

Srednicki solved a similar problem in the paper Entropy and Area: https://arxiv.org/pdf/hep-th/9303048.pdf, where he assumed an eigenfunction $f(x)_n=H_n(sqrt a x)$ where $H_n$ is the nth Hermite polynomial. I am wondering if a similar $f(x)_n$ exists in this case.