We review the higher-order supersymmetric quantum mechanics (H-SUSY QM), which involves differential intertwining operators of order greater than one. The iterations of first-order SUSY transformations are used to derive in a simple way the higher-order case. The second order technique is addressed directly, and through this approach unexpected possibilities for designing spectra are uncovered. The formalism is applied to the harmonic oscillator: the corresponding H-SUSY partner Hamiltonians are ruled by polynomial Heisenberg algebras which allow a straight construction of the coherent states.Comment: 42 pages, 12 eps figure

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.

For a class of Schrödinger Hamiltonians the supersymmetry transformations can degenerate to simple coordinate displacements. We examine this phenomenon and show that it distinguishes the Weierstrass potentials including the one-soliton wells and periodic Lamé functions. A supersymmetric sense of the addition formula for the Weierstrass functions is elucidated.

The confluent second-order supersymmetric quantum mechanics, with factorization energies ǫ 1 , ǫ 2 tending to a single ǫ-value, is studied. We show that the Wronskian formula remains valid if generalized eigenfunctions are taken as seed solutions. The confluent algorithm is used to generate SUSY partners of the Coulomb potential.

The confluent SUSY QM usually involves a second-order SUSY transformation where the two factorization energies converge to a single value. In order to achieve it, one generally needs to solve an indefinite integral, which limits the actual systems to which it can be applied. Nevertheless, not so long ago, an alternative method to achieve this transformation was developed through a Wronskian differential formula [Phys Lett. A 3756 (2012) 692]. In the present work, we consider the k-confluent SUSY transformation, where k factorization energies merge into a single value, and we develop a generalized Wronskian differential formula for this case. Furthermore, we explicitly work out general formulas for the third-and fourth-order cases and we present as examples the free particle and the single-gap Lamé potentials.

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In this article we will apply the first- and second-order supersymmetric quantum mechanics to obtain new exactly-solvable real potentials departing from the inverted oscillator potential. This system has some special properties; in particular, only very specific second-order transformations produce non-singular real potentials. It will be shown that these transformations turn out to be the so-called complex ones. Moreover, we will study the factorization method applied to the inverted oscillator and the algebraic structure of the new Hamiltonians.Comment: 19 pages, 8 figures, 2 tables. The new version has a new section for the algebras of the harmonic and inverted oscillators, a new appendix, and color figure

In these lecture notes we shall study first the supersymmetric quantum mechanics (SUSY QM), specially when applied to the harmonic and radial oscillators. In addition, we will define the polynomial Heisenberg algebras (PHA), and we will study the general systems ruled by them: for zero and first order we obtain the harmonic and radial oscillators, respectively; for second and third order PHA the potential is determined by solutions to Painlevé IV (PIV) and Painlevé V (PV) equations. Taking advantage of this connection, later on we will find solutions to PIV and PV equations expressed in terms of confluent hypergeometric functions. Furthermore, we will classify them into several solution hierarchies, according to the specific special functions they are connected with.