Laplace eigenfunction on a polygonal domain symmetric about an axis

Question

Consider a polygon $\Omega \subseteq \mathbb{R}^2$, and let us consider the usual Laplacian operator $\Delta = \partial_x^2 + \partial_y^2$, with Dirichlet boundary conditions. My question comes from trying to understand certain things in Chavel's book "Eigenvalues in Riemannian geometry". Suppose $\phi_2$ is a Dirichlet eigenfunction corresponding to the second eigenvalue $\lambda_2$ of $\Omega$. Now, let us reflect $\Omega$ about one of its sides, resulting in a polygon $\Omega^*$ that is symmetric about some axis. The extension of $\phi_2$ via reflection, which we call $\phi_2^*$, is clearly still a Dirichlet eigenfunction on $\Omega^*$.

My question: is there any way to understand what Dirichlet eigenvalue of $\Omega^*$ the extension $\phi^*_2$ corresponds to? My guess is that it should one of the low eigenvalues, but cannot prove this rigorously. Does it help to know whether $\lambda_2$ is non-repeated or not? Any help will be appreciated, thanks!

Remark: I asked this on MSE, but not getting any comments there, and I am now thinking that this question is probably more suited to MO. Please excuse the repetition.

Answer 1

There is no uniform bound, independent of the domain. Consider a rectangle with side lengths $\pi, \pi/K$, with $K\gg 1$. The eigenvalues are $m^2+n^2K^2$, $m,n\ge 1$. In particular, $\lambda_2$ is obtained for $m=2$, $n=1$.

If we now reflect about the long side, then the eigenvalues of the doubled rectangle are $m^2+n^2K^2/4$, and $2^2+K^2$ is no longer a low eigenvalue (it is about the $K$th eigenvalue).