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?



Diagonalise self-adjoint operator explicitly?
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.


