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

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