On a family of differential operators with the coupling parameter in the boundary condition

We study a family of differential operators $L_{\alpha }$ in two variables, depending on the coupling parameter $\alpha \ge 0$ that appears only in the boundary conditions. Our main concern is the spectral properties of $L_{\alpha }$, which turn out to be quite different for $\alpha \le 1$ and for $\alpha >1$. In particular, $L_{\alpha }$ has a unique self-adjoint realization for $\alpha \le 1$ and many such realizations for $\alpha >1$. In the more difficult case $\alpha >1$ an analysis of non-elliptic pseudodifferential operators in dimension one is involved.