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Exact analytical solutions for the bound states of a graphene Dirac electron in various magnetic fields with translational symmetry are obtained. In order to solve the time-independent Dirac-Weyl equation the factorization method used in supersymmetric quantum mechanics is adapted to this problem. The behavior of the discrete spectrum, probability and current densities are discussed.

We study a planar model of a non-relativistic electron in periodic magnetic and electric fields that produce a 1D crystal for two spin components separated by a half-period spacing. We fit the fields to create a self-isospectral pair of finite-gap associated Lamé equations shifted for a half-period, and show that the system obtained is characterized by a new type of supersymmetry. It is a special nonlinear supersymmetry generated by three commuting integrals of motion, related to the parityodd operator of the associated Lax pair, that coherently reflects the band structure and all its peculiarities. In the infinite period limit it provides an unusual picture of supersymmetry breaking.

The technique of differential intertwining operators (or Darboux transformation operators) is systematically applied to the one-dimensional Dirac equation. The following aspects are investigated: factorization of a polynomial of Dirac Hamiltonians, quadratic supersymmetry, closed extension of transformation operators, chains of transformations, and finally particular cases of pseudoscalar and scalar potentials. The method is widely illustrated by numerous examples

Polynomial deformations of the Heisenberg algebra are studied in detail. Natural realizations of them are given by the higher order susy partners 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. Indeed, it will be proved that the general systems ruled by such a kind of algebras, in the quadratic and cubic cases, involve Painlevé transcendents of type IV and V, respectively.

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.

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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.