I want to find out the eigenvalues of the following matrix
\begin{equation} \begin{pmatrix} e^{2k+J(p_x+p_y)} & e^{Jp_x} & e^{Jp_y} & e^{-2k-J(p_x+p_y)}\\ e^{Jp_x} & e^{2k+J(p_x-p_y)} & e^{-2k} & e^{-Jp_y} \\ e^{Jp_y} & e^{-2k}&e^{2k+J(p_y-p_x)} & e^{-Jp_x} \\ e^{-2k-J(p_y+p_x)} & e^{-Jp_y} & e^{-Jp_x} & e^{2k-J(p_x+p_y)}\\ \end{pmatrix} \end{equation}
I can go by brute force. In fact, I calculated the subdeterminants, but the resulting expressing is too lengthy, and the factorization is not obvious at all. Now Shur's decomposition is outside the window because, by default, A and D are non-invertible matrices. Any idea How I could evaluate fast the eigenvalues?
Thank you in advance