Solvability of a class of PT-symmetric non-Hermitian Hamiltonians: Bethe ansatz method

We use the Bethe ansatz method to investigate the Schrödinger equation for a class of PT-symmetric non-Hermitian Hamiltonians. Elementary exact solutions for the eigenvalues and the corresponding wave functions are obtained in terms of the roots of a set of algebraic equations. Also, it is shown that the problems possess sl(2) hidden symmetry and then the exact solutions of the problems are obtained by employing the representation theory of sl(2) Lie algebra. It is found that the results of the two methods are the same.