General conditions for the PT symmetry of supersymmetric partner potentials

Complete text of publication follows. A common feature of symmetries of quantum systems is that they restrict the form of the Hamiltonian, and consequently they also influence the structure of the energy spectrum. This is also the case with two symmetry concepts that are typically applied in non-relativistic quantum mechanics: supersymmetric quantum mechanics (SUSYQM) and PT symmetry. SUSYQM connects one-dimensional potentials pairwise via the relation V (±) (x) W2(x) ± dW/dx + ε, where ε is the factorization energy, V (-)(x) and V (+)(x) are the SUSY partner potentials, while W(x) is the superpotential. In the simplest case, when supersymmetry is unbroken, W(x) is defined in terms of the ground-state wavefunction of V (-)(x) as W(x) = - d/dx lnψ0(-)(x), and the factorization energy is chosen as ε E0(-). Under these conditions the SUSY partner potentials possess the same energy levels, except that E0(-) is missing from the spectrum of V (+)(x), and the degenerate levels are connected by the SUSY ladder operators A = d/dx + W(x) and A† = - d/dx + W(x). The PT symmetry of a Hamiltonian prescribes its invariance under simultaneous space and time inversion, which boils down to the condition V (x) = V*(-x) in the case of one-dimensional potentials. The unusual feature of this new symmetry concept is that PT-symmetric potentials are complex in general, nevertheless, they possess real energy eigen-values, unless PT symmetry is spontaneously broken, in which case the energy spectrum consists of complex conjugate energy pairs. The interplay of these two symmetry concepts has been analyzed in a number of works, and it has been found that when V(-)(x) has unbroken PT symmetry, then the same applies to V(+)(x), while the spontaneous breakdown of the PT symmetry of V (-)(x) implies the manifest breakdown of the PT symmetry of V (+) (x). The factorization energy ε was found to be real in the former case, and imaginary in the latter one. The examples analyzed to date, however, were restricted to some well-known exactly solvable potentials, which might not reveal some aspects of general situation. In order to gain a deeper insight, we separated the superpotential functions into real(R)/imaginary(I) and even(e)/odd(o) components W(x) = WRe(x) + WRo(x) + i WIe(x) + i WIo(x) and implemented the condition for PT symmetry of V(-)(x). This resulted in an inhomogeneous system of linear first-order differential equations for WRe(x) and WIo(x): W'Re - 2WRoWRe + 2WIeWIo = 0 W'Io - 2WIeWRe - 2WRoWIo Im(ε), which has two specific properties: the inhomogeneity is represented by a constant, Im(ε); the coefficients in the two equations are the same: WRo(x) and WIe(x). Once WRo(x) and WIe(x) is selected, the solution of this system can be given in a straightforward way. It is also clear that in the presence of a real factorization energy the inhomogeneous system reduces to an homogeneous one. In the inhomogeneous system was solved, while in the general solutions were given with illustrative examples for unbroken and spontaneously broken PT symmetry. It was found that in general the SUSY partner potential V(+)(x) can be PT-symmetric only for the trivial solution WRe(x) = WIo(x) = 0. (author)