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Consider the following homogeneous boundary value problem for a function/potential $u(x,y)$ on the infinite strip $[-\infty,\infty]\times[0,\pi/4]$ w/positive periodic coefficient/nductivity $\gamma(x+1,y)=\gamma(x,y)>0$: $$\begin{cases} \operatorname{div}(\gamma\nabla(u))=0,\\ u(x+1,y)=\mu u(x,y),\\ u_y(x,y+\pi/4)=0,\\ \gamma u_y(x,0)=-\lambda u(x,0). \end{cases}$$

In the uniform medium $\gamma\equiv const$, $\Delta u_k=0$ and the solution is of the separable form: $$u_k(x,y)=\mu_k^x(a\cos(\lambda_k y)+b\sin(\lambda_ky)),$$ $\lambda_k=|k|$ for an integer $k\in Z$ and, therefore, $$\lambda_k=|\log\mu_k|=|k|\beta(|k|)>0,$$ where $\beta\equiv 1.$ Let $z_k=|\log\mu_k|$. The analysis of the reflected/levant finite-difference problem suggests for any positive periodic $\gamma(x,y)>0$: $$\lambda_k=z_k\beta(z_k)>0,$$ for an analytic function $\beta(z)$, such that $\Re\beta(z)>0$ for $\Re(z)>0$ or equivalently: $$\beta(z)=z\int_{-\infty}^\infty\frac{(1+t^2)d\sigma(t)}{z-it}$$ for positive measure of bounded variation $\sigma$ on $[-\infty,\infty]$. In particular $\lambda_k$'s are not negative.

Any intuition for the existence and formula for $\beta(z)$? Does it follow from maximum principle for subharmonic functions?

The second part of the question is to replace positive coefficient $\gamma(x,y)>0$ w/separable $\alpha(y)\delta(x)$, such that $\beta_{\alpha\delta}=\beta_\gamma$...

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