Complex square well—a new exactly solvable quantum mechanical model

Recently, a class of PT -invariant quantum mechanical models described by the nonHermitian HamiltonianH = p2 + x2(ix) was studied. It was found that the energy levels for this theory are real for all > 0. Here, the limit as →∞ is examined. It is shown that in this limit, the theory becomes exactly solvable. A generalization of this Hamiltonian, H = p2 + x2M(ix) (M = 1, 2, 3, . . .) is also studied, and this PT -symmetric Hamiltonian becomes exactly solvable in the largelimit 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 largelimit are obtained.