L M Nieto

Self-isospectrality, special supersymmetry, and their effect on the band structure.

We study a planar model of a nonrelativistic 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 parity-odd 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.

Second-order supersymmetric periodic potentials

Abstract Irreducible second-order SUSY transformations are applied to periodic Hamiltonians in order to find physically acceptable partner potentials with the same band structure as the initial one. Lames potentials are analized in the same context. The main differences with the SUSY approach to potentials allowing for a discrete spectrum are also discussed.

The finite difference algorithm for higher order supersymmetry

Abstract The higher order supersymmetric partners of the Schrodingers Hamiltonians can be explicitly constructed by iterating a simple finite difference equation corresponding to the Backlund transformation. The method can completely replace the Crum determinants. Its limiting, differential case offers some new operational advantages.

Exact analytic solutions for a Dirac electron moving in graphene under magnetic fields

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.

The supersymmetric modified Pöschl-Teller and delta well potentials

New supersymmetric (SUSY) partners of the modified Poschl-Teller and the Diracs delta well potentials are constructed in closed form. The resulting one-parametric potentials are shown to be interrelated by a limiting process. The range of values of the parameters for which these potentials are free of singularities is exactly determined. The construction of higher-order SUSY partner potentials is also investigated.

Intertwining technique for the one-dimensional stationary Dirac equation

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.

A study of the bound states for square potential wells with position-dependent mass

A potential well with position-dependent mass is studied for bound states. Applying appropriate matching conditions, a transcendental equation is derived for the energy eigenvalues. Numerical results are presented graphically and the variation of the energy of the bound states are calculated as a function of the well-width and mass.

Shape-invariant quantum Hamiltonian with position-dependent effective mass through second-order supersymmetry

A second-order supersymmetric approach is taken to the system describing motion of a quantum particle in a potential endowed with position-dependent effective mass. It is shown that the intertwining relations between second-order partner Hamiltonians may be exploited to obtain a simple shape-invariant condition. Indeed, a novel relation between potential and mass functions is derived, which leads to a class of exactly solvable models. As an illustration of our procedure, two examples are given for which one obtains whole spectra algebraically. Both shape-invariant potentials exhibit harmonic-oscillator-like or singular-oscillator-like spectra depending on the values of the shape-invariant parameter.

Polynomial Heisenberg algebras

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.

New isospectral oscillator potentials

Abstract Super-supersymmetric quantum mechanics arises if a second-order differential operator A † intertwines two different Hamiltonians H, H , i.e. H A † = A † H . This technique is used to generate a 2-parameter family of strictly isospectral oscillator potentials embracing the Abraham-Moses case. As a by-product, a 1-parameter family of potentials isospectral to the oscillator, except for the level of the first excited state, is derived.