Non-Hermitian supersymmetry and singular, P T-symmetrized oscillators

Hermitian supersymmetric partnership between singular potentials V(q) = q² + G/q² breaks down and can only be restored on certain ad hoc subspaces (Das A and Pernice S 1999 Nucl. Phys. B 561 357). We show that within extended, P T -symmetric quantum mechanics, the supersymmetry between singular oscillators can be completely re-established in a way which is continuous near G = 0 and leads to a new form of the bosonic creation and annihilation operators.