On a certain problem, I obtain $U(r, \theta) = T_s(\theta) J_s(\sqrt{\lambda}r)$, where $J_s$ is the Bessel's function of order $s$, and we know by the Dirichlet boundary conditions that $U(r_0, \theta)=0$. As $T_s=a_s \cos(s \theta) + b_s \sin (s \theta) \not= 0$, with $\theta \in [0, 2 \pi]$, then $J_s(\sqrt{\lambda} r_0)=0$. Hence $\sqrt{\lambda} r_0$ is a zero of the Bessel's function of order $s$.
How can we extract the eigenvalues of the eigenfunctions $U$?