The progress of the factorization method since the 1935 work of Dirac is briefly reviewed. Though linked with older mathematical theories the factorization seems an autonomous 'driving force', offering substantial support to the present day Darboux and Bäcklund approaches.

The higher order supersymmetric partners of the Schroedinger's Hamiltonians can be explicitly constructed by iterating a simple finite difference equation corresponding to the Baecklund transformation. The method can completely replace the Crum determinants. Its limiting, differential case offers some new operational advantages.Comment: LaTeX, 12 pages, 3 figures. To appear in Phys. Lett.

A new form to construct complex superpotentials that produce real energy spectra in supersymmetric quantum mechanics is presented. This is based on the relation between the nonlinear Ermakov equation and a second order differential equation of the Schrödinger type. The superpotentials so constructed are characterized by the Ermakov parameters in such a way that they are always complex-valued and lead to non-Hermitian Hamiltonians with real spectra, whose eigenfunctions form a bi-orthogonal system. As applications we present new complex supersymmetric partners of the free particle that are PT -symmetric and can be either periodic or regular (of the Pöschl-Teller form). A new family of complex oscillators with real frequencies that have the energies of the harmonic oscillator plus an additional real eigenvalue is introduced.

The dynamical algebra associated to a family of isospectral oscillator Hamiltonians is studied through the analysis of its representation in the basis of energy eigenstates. It is shown that this representation becomes similar to that of the standard Heisenberg algebra, and it is dependent of a parameter w ≥ 0. We name it distorted Heisenberg algebra, where w is the distortion parameter. The corresponding coherent states for an arbitrary w are derived, and some particular examples are discussed in full detail. A prescription to produce the squeezing, by adequately selecting the initial state of the system, is given.