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 $\alpha\ge0$ 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.