Abstract: In this talk, we consider the one-dimensional Schr\"odinger operators with the periodic $\delta^{(1)}$-interactions and discuss its spectrum. The perturbation of the free Hamiltonian by the distributions $\delta^{(1)}$ is defined through the distribution theory for the discontinuous test functions. It turns out by the Floquet--Bloch theory that the spectrum of the above operators has the band structure. In order to analyze the spectrum more precisely, we discuss the coexistence problem. Namely, we determine whether the $j$th spectral gap is degenerate or not for each $j\in{\bf N}$. We consider the proof of this problem by using the rotation number.