The relation of supersymmetric quantum mechanics (SUSYQM) is discussed with two other symmetry-based approaches: PT symmetry and a differential realization of the su(1, 1) and su(2) algebra. It is demonstrated that PT symmetry imposes conditions on the even and odd parts of the real and imaginary components of the superpotential W(x), and these are expressed in terms of a system of first-order linear differential equations, which is homogeneous when the factorization energy is real and inhomogeneous when it is complex. The formal solution of this system is presented for various special cases as well as for the general case. It is shown that a trivial solution of this system corresponds to the unbroken PT symmetry for the Scarf II potential. The formalism of SUSYQM is also linked with that of the potential algebra approach, and it is demonstrated that the J+ and J- ladder operators of some su(1,1) or su(2) algebras act on series of degenerate levels of different potentials essentially in the same way as the A and At shift operators of SUSYQM. Examples are presented for su(1, 1) and su(2) potential algebras, as well as for spectrum generating algebras of the same type. Possible generalizations of this construction are also pointed out.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)