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=p2+x2(ix)ε was studied. It was found that the energy levels for this theory are real for all ε≥0. Here, the limit as ε→∞ is examined. It is shown that in this limit, the theory becomes exactly solvable. A generalization of this Hamiltonian, H=p2+x2M(ix)ε (M=1,2,3, ... ) is also studied, and this PT-symmetric Hamiltonian becomes exactly solvable in the large-ε 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-ε limit are obtained. (author)