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

摘要

We study a family of differential operators Lα 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 Lα, which turn out to be quite different for α<1 and for α>1. In particular, Lα 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.