# Particle in a ring with boundary terms at the origin

Suppose we have a particle in a ring of N total lattice sites, with lattice spacing $$a=1$$. The ring is supposed to have a "boundary" at lattice site $$n=0$$, where the hopping is different. The bulk wavefunctions, away from the boundary satisfy: $$\begin{eqnarray} E\psi_{n}=-t\psi_{n-1}-t\psi_{n+1} \end{eqnarray}$$ from which solutions of the form: $$\begin{eqnarray} \psi_{n}=Ae^{ikn}+Be^{-ikn} \end{eqnarray}$$ can be shown to satisfy the equations provided $$E=-2t\cos(k)$$. However, close to the boundary $$n=0$$, at lattice sites $$n=1,N-1$$, we have: $$\begin{eqnarray} (E+a)\psi_{1}=-t\psi_{2}+\beta\psi_{N-1}\nonumber\\ (E+a)\psi_{N-1}=-t\psi_{N-2}+\beta\psi_{1} \end{eqnarray}$$ If we substitute now the ansatz above for $$\psi_{n}$$, there are two equations to determine $$A,B$$, which are complex amplitudes. Another condition must be fullfilled by the normalization of the wavefunction, which in turn determines the magnitude of $$A,B$$, i.e. $$|A|,|B|$$. But when it comes to determine the phases, I am a bit confused, because the equations seem to be the same by just changing $$k\to -k$$. This is because due to PBC of the ring, we must have: $$\begin{eqnarray} \psi_{n+N}=\psi_{n}\to e^{\pm ikN}=1\to k=\frac{2\pi n}{N},n\in\mathbb{Z} \end{eqnarray}$$ Using the PBC the equations become: $$\begin{eqnarray} A\left[(E+a)e^{ik} + te^{i2k} + \beta e^{-ik}\right]=-B\left[(E+a)e^{-ik} + te^{-i2k} + \beta e^{ik}\right]\\ A\left[(E+a)e^{-ik} + te^{-i2k} + \beta e^{ik}\right]=-B\left[(E+a)e^{ik} + te^{i2k} + \beta e^{-ik}\right] \end{eqnarray}$$ Satisfying both equations simultaneously seems contradictory to me; note that one is NOT the complex conjugate of the other, since that would require $$A\to A^{*}$$. Therefore, I don't know how to interpret these equations. It is also clear that the ratio $$A/B$$ is a pure phase.

How does the physical significance of such solutions take place? Can we determine in advance the chirality of the particle?