Consider the spin^{c} structure on the flat standard 3-torus, which you get from the trivial (or any other) spin structure. Its associated vector bundle can be identified with a trivial bundle with fibre $\mathbb{C}^2$. spin^{c} Dirac operators on this bundle are parametrized by one-forms and look like $D_\alpha = D_0+ic_\alpha$, where $D_0$ is the spin Dirac operator and the $c$ means Clifford multiplication.

My aim is now to find a spectral decomposition for $D_\alpha$. If α is closed, this can be easily done by reducing everything to the case where α is harmonic. The case where α is not closed seems to be more tricky, so I would like to ask the community:

- What is the spectrum of $D_\alpha$?
- How can you construct eigenvectors?

Thank you!