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

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