**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.