The eigenvalue function of a family of Sturm–Liouville operators

We define a function $\mu^-(\gamma)$ in such a way that its value at every point $\gamma\in(-\infty,\pi)$, $\gamma=\beta-\pi n$, $\beta\in[0,\pi)$, $n=0,1,2,\dots$, coincides with an eigenvalue $\mu_n(\alpha,\beta)$ of the Sturm–Liouville problem $-y''+q(x)y=\mu y$, $y(0)\cos\alpha+y'(0)\sin\alpha=0$, $y(\pi)\cos\beta+y'(\pi)\sin\beta=0$ (for some $\alpha\,{\in}\,(0,\pi]$). We find necessary and sufficient conditions for a function to have this property for a real $q\in L^1[0,\pi]$.