Complex square well - a new exactly solvable quantum mechanical model

Recently, a class of PT-invariant quantum mechanical models described by the non-Hermitian Hamiltonian H = p(2) + x(2)(ix)(epsilon) was studied It was found that the energy levels for this theory are real for all epsilon greater than or equal to 0. Here, the limit as epsilon --> infinity is examined. It is shown that in this limit, the theory becomes exactly solvable. A generalization of this Hamiltonian, H = p(2) + x(2M)(ix)(epsilon) (M = 1, 2, 3,...) is also studied, and this PT-symmetric Hamiltonian becomes exactly solvable in the large-epsilon limit as well. In effect, what is obtained in each case is a complex analogue of the Hamiltonian for the square-well potential. Expansions about the large-epsilon limit are obtained.