along the same lines as that of the $N=$even case, and will not be given here.

Single-particle states with positive energy
===========================================

The equivalence between the uniform ring and the SSD problem is established only for the ground states of many fermions. Here, we comment on single-particle excited states above the Fermi level, i.e., the positive energy ($\xi >0$) solutions of the SSD problem. From the correspondence between the SSD and inverse sine-square problems, one can see that they translate into the eigenfunctions of the attractive $1/\sin^2$ potential problem. As noted in Appendix A, the solution of the $1/\sin^2$ potential problem with the attractive coupling is classified into two cases: (i) $0<\xi <\frac{1}{4} (\frac{\pi}{L})^2 $ and (ii) $\frac{1}{4} (\frac{\pi}{L})^2 <\xi$. The case (i) corresponds to Eq. (\[SSDpotential\]) with $0<\beta<1$. For a non-integer value of $\beta$, however, the solution of Eq. (\[SSDpotential\]) is given by an analytic continuation of the Gegenbauer polynomial, which exhibits a singular behavior at the boundaries. Thus, the physical solution is permitted only in the limit of $\beta\to 0$ or $1$, which is reduced to the plane-wave solution of $\beta=0$ or 1. Recalling that the boundary condition for $\beta=0$ is Neumann and that for $\beta=1$ is the 