An exactly solvable symmetric potential from the Natanzon class

Abstract

The ${\cal PT}$ symmetric version of the generalised Ginocchio potential, a member of the general exactly solvable Natanzon potential class is analysed and its properties are compared with those of ${\cal PT}$ symmetric potentials from the more restricted shape-invariant class. It is found that the ${\cal PT}$ symmetric generalised Ginocchio potential has a number of properties in common with the latter potentials: it can be generated by an imaginary coordinate shift $x\to x+{\rm i}\epsilon$; its states are characterised by the quasi-parity quantum number; the spontaneous breakdown of ${\cal PT}$ symmetry occurs at the same time for all the energy levels; and it has two supersymmetric partners which cease to be ${\cal PT}$ symmetric when the ${\cal PT}$ symmetry of the original potential is spontaneously broken.