Quantum Integrability of Certain Boundary Conditions

We study the quantum integrability of the O(N) nonlinear $\sigma$ (nls) model and the O(N) Gross-Neveu (GN) model on the half-line. We show that the \nls model is integrable with Neumann, Dirichlet and a mixed boundary condition, and that the GN model is integrable if ${\psi }_{+}^a\text{x}=\pm {\psi }_{-}^a\text{x}$. We also comment on the boundary condition found by Corrigan and Sheng for the O(3) nls model.