A self-adjoint “simultaneous crossing of the axis”

By constructing the corresponding Green's function in a trapezoidal domain, we establish the existence of self-adjoint realizations of$$A = \frac{{\partial ^2 }}{{\partial t^2 }} - \frac{{\partial ^2 }}{{\partial s^2 }}$$ incorporating boundary conditions of the formu(s, 0)=u(s, T)=0. Such operators correspond to the historically important concept of a “simultaneous crossing of the axis” for vibrating strings.