On spectrum of periodic boundary value problem

Question

Consider the following boundary value problem on the infinite strip $(-\infty,\infty)\times[0,1]$ w/periodic conductivity $\gamma(x,y)=\gamma(x+2\pi,y)>0$: $$\begin{cases} \operatorname{div}(\gamma\nabla u)=(\gamma u_x)_x+(\gamma u_y)_y=0,\\ u(x+2\pi,y)=\mu u(x,y),\\ u(x,1)=0,\\ u(x,0)=0 \mbox{ or } u_y(x,0)=0. \end{cases}$$ Let's denote the set of possible values of the eigenvalue parameter $\mu$, for which the system has a solution $\mu_\gamma$. Does there exist a conductivity $\alpha(y)$ on the strip, depending only on $y$, such that $$\mu_\gamma=\mu_\alpha?$$ The answer is "yes" for the finite-difference version of the problem, but the proof, that I've got doesn't apply to the continuous case...