Functional Methods for the Solution of One-Dimensional Quantum Systems
The framework of the Quantum Inverse Scattering Method is used to study the hamiltonian of the XXX and XXZ spin chain with general boundary fields. Key ingredient is the underlying algebraic structure which is a combination of the Yang-Baxter algebra and a so-called Reflection algebra including boundary fields of arbitrary direction and strength. For spin chains with diagonal boundary fields this setup has been well studied using algebraic Bethe ansatz and the inverse problem was solved by Kitanine for infinite chain lengths. These results are picked up and generalized to arbitrary lengths using non-linear integral equations. In the case of non-diagonal boundary fields the lack of a reference state or pseudo vacuum prohibits the solution by algebraic Bethe ansatz. The method of separation of variables is not constrained in that sense and is applied to the XXX chain and a spin-boson model. Finally a different approach to the case of non-diagonal boundary conditions is studied. Starting from the so-called fusion hierarchy non-linear integral equations are derived bearing the possibility to extract information about an eigenvalue of a specific state.