sp.spectral theory - $Spin^c$-Dirac-operator on the 3-torus

Here is a possible unpromising start that hints at probable headaches. Square the Dirac to get

$$ D_alpha^2= Delta+ c(dalpha)$$

where $c(dalpha)$ denotes the Clifford multiplication by the $2$-form $dalpha$. Note that

$$ {rm spec}(D_alpha^2)= bigl(; mathrm{spec}(D_alpha);bigr)^2 $$

To find ${rm spec}(D_alpha^2)$ you need to understand spectrum of ordinary differential operators of the form

$$ -partial^2_theta + A(theta) $$

acting on functions $u: S^1 to mathbb{C}^2$ where $A(theta)$ is a $2times 2$ complex hermitian matrix depending smoothly on $thetain S^1$. I don't know how to find the spectrum of such an operator but maybe you can find something in the literature.