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 Backlund approaches.

http://homepage.cem.itesm.mx/fdelgado/ciencia/cadi/ref06.pdf

Factorization: little or great algorithm?

Using an iterative construction of the first-order intertwining technique, we find k- parametric families of exactly solvable anharmonic oscillators whose spectra consist of a part isospectral to the oscillator plus k additional levels at arbitrary positions below E0D 1 . It is seen that the 'natural' ladder operators for these systems give place to polynomial nonlinear algebras, and it is shown that these algebras can be linearized. The coherent states construction is performed in the nonlinear and linearized cases.

/pdf/higher-order-susy-linearized-nonlinear-heisenberg-algebras-5fmtxs24z8.pdf

Higher-order SUSY, linearized nonlinear Heisenberg algebras and coherent states

We construct a displacement-operator-type nonlinear coherent state and examine some of its properties. In particular, it is shown that this nonlinear coherent state exhibits nonclassical properties such as squeezing and sub-Poissonian behaviour.

https://iopscience.iop.org/article/10.1088/1464-4266/2/1/311/pdf

New nonlinear coherent states and some of their nonclassical properties

Polynomial deformations of the Heisenberg algebra are studied in detail. Some of their natural realizations are given by the higher order susy partners (and not only by those of first order, as is already known) of the harmonic oscillator for even-order polynomials. Here, it is shown that the susy partners of the radial oscillator play a similar role when the order of the polynomial is odd. Moreover, it will be proved that the general systems ruled by such kinds of algebras, in the quadratic and cubic cases, involve Painleve transcendents of types IV and V, respectively.

https://www.fis.cinvestav.mx/~david/papers/polyheisalg04.pdf

Polynomial Heisenberg algebras

Coherent states are derived for one-dimensional systems generated by supersymmetry from an initial Hamiltonian with a purely discrete spectrum for which the levels depend analytically on their subindex. It is shown that the algebra of the initial system is inherited by its SUSY partners in the subspace associated to the isospectral part or the spectrum. The technique is applied to the harmonic oscillator, infinite well and trigonometric Poeschl-Teller potentials.

Coherent states for Hamiltonians generated by supersymmetry

The dynamical algebra associated with 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 on a parameter omega >or=0. We call it the distorted Heisenberg algebra, where omega is the distortion parameter. The corresponding coherent states for an arbitrary omega are derived, and some particular examples are discussed in detail. A prescription to produce the squeezing, by adequately selecting the initial state of the system, is given.

C D J Fernandez

Papers

Distorted Heisenberg algebra and coherent states for isospectral oscillator Hamiltonians