Consider the linear constant coefficient differential operator $P$ on the Hilbert space $L^2([0,1]^2;\mathbb C^2)$ $$P= \begin{pmatrix} D_{z}+c & a \\ b & D_{z}+c \end{pmatrix}$$ where $D_z=-i \partial_z =- i(\partial_{x_1} -i \partial_{x_2}).$
Here, $a,b,c$ are just some complex numbers.
I wonder whether one can explicitly compute the spectrum of the self-adjoint operator $P^*P$, with periodic boundary conditions, then?
By noting that $-i\partial_{x_1} $ is diagonalized by $e^{ik_1 x_1} $ and $-i\partial_{x_2} $ by $e^{ik_2 x_2} $, the problem reduces to a $2\times 2$ diagonalization for each $(k_1,k_2)$-block. The resulting eigenvalues are (denoting $k=k_1-ik_2 $) $$ \frac{1}{2} (|a|^2 + |b|^2 ) +|k+c|^2 \pm\frac{1}{2} \sqrt{(|a|^2 -|b|^2 )^2 +4(|a|^2 + |b|^2 )|k+c|^2 +8\,\mbox{Re}\, (a^*b^*(k+c)^2)} $$ where $k_1 , k_2 \in 2\pi \mathbb{Z} $ in view of the boundary conditions.
