The finite PT-symmetric square well potential

The -symmetric version of the one-dimensional finite square well potential is constructed by supplementing the real finite square well potential with constant imaginary components outside the well. This study is motivated by the unusual features of the -symmetric Rosen-Morse II potential, which has an imaginary component with similar asymptotics. The exponentially vanishing (i.e. normalizable) solutions are constructed, after determining the corresponding energy eigenvalues from the zeros of a transcendental equation. It is found that only real energy eigenvalues are allowed, similarly to the case of the Rosen-Morse II potential. Transmission and reflection coefficients are determined and are found to exhibit handedness. It is shown that due to the non-vanishing imaginary potential component, bound states correspond to the zeros of the reflection coefficient. Similarities and differences with respect to the real finite square well and the -symmetric Rosen-Morse II potential are discussed.