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Trends in Supersymmetric Quantum Mechanics

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Integrability, Supersymmetry and Coherent States

Part of the book series: CRM Series in Mathematical Physics ((CRM))

Abstract

Along the years, supersymmetric quantum mechanics (SUSY QM) has been used for studying solvable quantum potentials. It is the simplest method to build Hamiltonians with prescribed spectra in the spectral design. The key is to pair two Hamiltonians through a finite order differential operator. Some related subjects can be simply analyzed, as the algebras ruling both Hamiltonians and the associated coherent states. The technique has been applied also to periodic potentials, where the spectra consist of allowed and forbidden energy bands. In addition, a link with non-linear second-order differential equations, and the possibility of generating some solutions, can be explored. Recent applications concern the study of Dirac electrons in graphene placed either in electric or magnetic fields, and the analysis of optical systems whose relevant equations are the same as those of SUSY QM. These issues will be reviewed briefly in this paper, trying to identify the most important subjects explored currently in the literature.

Dedicated to my dear friend and colleague Véronique Hussin.

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Notes

  1. 1.

    We choose here a notation consistent with Sect. 2. Please do not confuse the intertwining operators of Eq. (116) with the magnetic field B, its magnitude \(\mathcal{B}(x)\), or any of its components.

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Acknowledgements

The author acknowledges the financial support of the Spanish MINECO (project MTM2014-57129-C2-1-P) and Junta de Castilla y León (VA057U16).

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  1. Departamento de Física, Cinvestav, Ciudad de México, Mexico

    David J. Fernández C.

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  1. Department of Physics, Ankara University, Ankara, Turkey

    Şengül Kuru

  2. Department of Theoretical Physics, Atomic Physics and Optics, University of Valladolid, Valladolid, Spain

    Javier Negro

  3. Department of Theoretical Physics, Atomic Physics and Optics, University of Valladolid, Valladolid, Spain

    Luis M. Nieto

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C., D.J.F. (2019). Trends in Supersymmetric Quantum Mechanics. In: Kuru, Ş., Negro, J., Nieto, L. (eds) Integrability, Supersymmetry and Coherent States. CRM Series in Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-20087-9_2

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