The Dirac spectrum for $S^n$ is well known along with its multiplicities. In Appendix D of https://arxiv.org/pdf/1510.05663.pdf author computes the Dirac spectrum of $RP^4$ from that of $S^4$. The argument author uses is that $RP^4$ has two $pin^+$ structures and applying parity condition on Dirac spectrum of $S^4$ gives the spectrum for $RP^4$. My question is
1. Why parity condition and how does it relate pin and spin structures?
2. Suppose, I want to obtain the Dirac spectrum for $RP^n$. I know n= 1 is easy as it has two inequivalent spin structures same as $S^1$. But how do I obtain it for $n =2, 3, 5$ etc. I also know $RP^3$ is orientable (for odd n) and has 2 $pin^+$ and 2 $pin^-$ structures. But how is its spectrum different from that of $S^3$?