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 ${\mathrm{\Omega }}_s\subset {\mathbb{R}}^{N+1}$ with $N\ge 1$, where \begin{equation} Ω_s=\left{(x,t)\in $\mathbb{R}$^N\times $\mathbb{R}$:\vert x\vert<1+s\cos \left(\frac{2π}{T(s)}t\right)+s w_s\left(\frac{2π}{T(s)}t\right)\right},\nonumber \end{equation} such that \begin{equation} -Δu=λu,, $\text{in}$,,Ω, ,, \partial_νu=0,,,u=$\text{const}$,,$\text{on}$,,\partialΩ\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 “$\mathrm{\partial }\mathrm{\Omega }$ 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.