A week ago I became really surprised to realize that there are planar rings (annuli) for which there is an eigenvalue of the Dirichlet Laplacian with the multiplicity at least 3. I was quite sure that the situation for rings is the same as for the disk, where the multiplicity of any eigenvalue is either 1 or 2. The latter result is a consequence of Bourget’s hypothesis which states that no Bessel functions $J_\nu$ and $J_{\nu+m}$ with natural $\nu, m$ have common positive zeros. Moreover, in the well-known survey article of Grebenkov and Nguen 1, Section 3.2 (in between (3.7) and (3.8)), it is written that eigenvalues (in rings) are either simple or twice degenerate. This assertion appears to be not true.
So, let us consider the ring $A_r := \{x \in \mathbb{R}^2: r<|x|<1\}$ and the eigenvalue problem
\[\left\{ \begin{aligned} -\Delta u &= \lambda u &&{\rm in}\ A_r, \\ u&=0 &&{\rm on }\ \partial A_r, \end{aligned} \right.\]-
Grebenkov, D. S., & Nguyen, B. T. (2013). Geometrical structure of Laplacian eigenfunctions. SIAM Review, 55(4), 601-667. ↩