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 $k \in 2\pi \mathbb{Z}.$