Sign-changing solutions to Schiffer's overdetermined problem on wavy cylinder

In this paper, we prove the existence of $k$ families of smooth unbounded domains $\Omega_s\subset\mathbb{R}^{N+1}$ with $N\geq1$, where \begin{equation} \Omega_s=\left\{(x,t)\in \mathbb{R}^N\times \mathbb{R}:\vert x\vert<1+s\cos \left(\frac{2\pi}{T(s)}t\right)+s w_s\left(\frac{2\pi}{T(s)}t\right)\right\},\nonumber \end{equation} such that \begin{equation} -\Delta u=\lambda u\,\, \text{in}\,\,\Omega, \,\, \partial_\nu u=0,\,\,u=\text{const}\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} admits a bounded sign-changing solution with exactly $k+1$ nodal domains. These results can be regarded as counterexamples to the Schiffer conjecture on unbounded domain. These results also indicate that there exist non-spherical unbounded regions without Pompeiu property. Our construction shows that the condition "$\partial\Omega$ is homeomorphic to the unit sphere" is necessary for Williams conjecture to hold. In addition, these conclusions may have potential applications in remote sensing or CT.