Closed-form solutions of the generalized boundary-condition phase equation

Derive closed-form expressions for the phase variable \(\theta\) solving the generalized-boundary-condition trigonometric equation \(\sin((m+1)\theta)+A\sin(m\theta)-B\sin((m-1)\theta)-C\sin\theta=0\) for values of \(A\) other than the explicitly solvable cases \(A=0,\pm1\) considered in the paper, thereby completing the analytic determination of the corresponding eigenvalues and eigenstates.

Background

The generalized-boundary-condition analysis reduces the spectral problem to a trigonometric equation involving the system size $m$ and three real parameters $A$, $B$, and $C$, which are determined by the bulk and boundary hopping amplitudes. The phase $\theta$ controls the generalized Bloch factor and therefore determines the energy spectrum and eigenstates.

The paper obtains exact solutions for several special parameter choices, including $A=0,B=1$, $B=-1,C=0$, and $B=C=0$ with $A=0,\pm1$. It explicitly states that closed-form solutions are not obtained for other values of $A$, leaving the general analytic solution of the phase equation unresolved.