= Solution
The <quantum rotor> has orientation angle $\phi$ of period $2\pi$, so physical wavefunctions satisfy $\psi(\phi+2\pi)=\psi(\phi)$. The exponential solutions of $H u=Eu$ are therefore
$$
\boxed{u_m(\phi)=\frac{e^{im\phi}}{\sqrt{2\pi}},\qquad E_m=\frac{\hbar^2m^2}{2I},\qquad m\in\mathbb Z.}
$$
They are normalized and <orthogonal> on $[0,2\pi)$. The corresponding <angular momentum> operator is $L_z=-i\hbar\partial_\phi$, with <eigenvalue> $m\hbar$. The $m=0$ ground level is nondegenerate, and every $|m|>0$ <energy level> has the twofold degeneracy $m\leftrightarrow-m$. This uses ordinary periodic boundary conditions, with no flux or twisted <quantum phase>.
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