Consider the following Floquet eigenvalue equation: $$ \left(\frac{d^2}{dx^2}-k^2+\omega^2\ V(x)\right)\psi(x)=0 $$ where $V(x+L)=V(x)>0$ is a positive periodic potential, $k$ is a real parameter, $\omega$ is the eigenvalue, and the eigenfunction satisfies the periodic boundary condition $\psi(x+L)=\psi(x)$, $\frac{d}{dx}\psi(x+L)=\frac{d}{dx}\psi(x)$.
From the numerical solutions of several different periodic potentials $V(x)$, I find that all branches of eigenvalues approach to linear asymptotic dispersion $\omega(k)$ with respect to the parameter $k$, namely $\lim_{k\rightarrow\infty}d\omega/dk=\mathrm{constant}>0$, and it seems that the asymptotic slopes for different branches of eigenvalues are identical. The following figures show two examples. However, I cannot prove this conjecture. Can some one help me prove or disprove this conjecture?
The insets are the profiles of the potential function $V(x)$ in one period.