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?

• What do you mean by $\partial_{x_1}$ and $\partial_{x_2}$ on a space whose functions depend only on one real variable? – Jochen Glueck Jan 7 at 23:12
• @JochenGlueck sorry, should have been $[0,1]^2.$ – Sascha Jan 7 at 23:18
• You'd need to give some boundary conditions for this to be a well-defined question. – Michael Engelhardt Jan 8 at 0:46
• @sorry, yes, periodic ones. – Sascha Jan 8 at 0:47
• @MichaelEngelhardt thank you for your efforts – Sascha Jan 8 at 1:50

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.
• I am essentially doing a 2-dimensional Fourier transform, to the basis $e^{i(k_1 x_1 + k_2 x_2 )}$. That's a unitary transformation on $L^2$. As is the final $2\times 2$ diagonalization. – Michael Engelhardt Jan 8 at 4:53