Consider the 2D Dirac operator


$$H = \begin{pmatrix} 0 & \partial\_{\bar z} \\ \partial\_z & 0 \end{pmatrix}$$


where $\partial\_z = \partial\_x - i \partial\_y$ and $\partial\_{\bar z} = \partial\_x + i \partial\_y.$


This implies by using the Bloch transform that there exist functions $\psi\_{\lambda}$ such that


$$H \psi\_{\lambda}= \lambda \psi\_{\lambda}$$ and $\psi\_{\lambda}$ are periodic on $[0,1] \times [0,1]$, i.e. $\psi(x+1,y)=\psi(x,y)$ and $\psi(x,y+1)=\psi(x,y).$


Can we find these functions explicitly?


To give an example where the situation is somewhat easier.


For the second derivative on $\mathbb R$ we have that


$$-\frac{d^2}{dx^2} e^{ikx} = k^2 e^{ikx}$$


where $e^{ikx}$ is periodic on $[0,1]$ if $k \in 2\pi \mathbb{Z}.$



Periodic eigenfunctions for 2D Dirac operator
$$
\left( \begin{array}{c} 1 \\ \frac{k\_x - ik\_y }{\sqrt{k\_x^2 + k\_y^2 } } \end{array} \right) e^{i(k\_x x + k\_y y)} \ \ \ \mbox{with eigenvalue} \ \ \ i\sqrt{k\_x^2 + k\_y^2 }
$$
and
$$
\left( \begin{array}{c} 1 \\ -\frac{k\_x - ik\_y }{\sqrt{k\_x^2 + k\_y^2 } } \end{array} \right) e^{i(k\_x x + k\_y y)} \ \ \ \mbox{with eigenvalue} \ \ \ -i\sqrt{k\_x^2 + k\_y^2 }
$$
with $k\_x , k\_y \in 2\pi \mathbb{Z} $ for the desired periodicity.


