Quantum and classical solutions for free particle in wedge billiards

We have studied the quantum and classical solutions of a particle constrained to move inside a sector circular billiard with angle $\theta_w$ and its pacman complement with angle $2\pi-\theta_w$. In these billiards rotational invariance is broken and angular momentum is no longer a conserved quantum number. The "fractional" angular momentum quantum solutions are given in terms of Bessel functions of fractional order, with indices $\lambda_p={p\pi \over {\theta_w}}$, $p=1,2,...$ for the sector and $\mu_q={q\pi \over {2\pi - \theta_w}}$, $q=1,2...$ for the pacman. We derive a ``duality'' relation between both fractional indices given by $\lambda_p={{p\mu_q} \over {2\mu_q - q}}$ and $\mu_q = {{q\lambda_p} \over {2\lambda_p - p}}$. We find that the average of the angular momentum $\hat L_z$ is zero but the average of $\hat L^2_z$ has as eigenvalues $\lambda_p^2$ and $\mu_q^2$. We also make a connection of some classical solutions to their quantum wave eigenfunction counterparts.