We propose a new simplified procedure for finding the analytical solutions of the stationary one-dimensional Schrödinger equation, with asymmetric boundary conditions imposed on the equation.The essence of the method consists in expressing the general solution explicitly in terms of the boundary parameters in a form which, by itself, satisfies one of the boundary conditions involved; then, the other boundary condition gives straightforwardly the characteristic equation.This method may turn out to be beneficial with regard to the recently growing interest in one-dimensional quantum systems.
The integrable open-boundary conditions for the one-dimensional Bariev chain are considered. The diagonal boundary
K matrices are found and the commuting transfer matrix is constructed. Since the local monodromy matrix as well as the quantum
R matrix do not possess the crossing symmetry, our construction shows that Sklyanin's formalism may be extended to apply to any one-dimensional systems integrable by the quantum inverse scattering method.