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Trends in supersymmetric quantum mechanics. (English) Zbl 1425.81045
Kuru, Şengül (ed.) et al., Integrability, supersymmetry and coherent states. A volume in honour of Professor Véronique Hussin. In part selected contributions from the 6th international workshop on new challenges in quantum mechanics: integrability and supersymmetry, Valladolid, Spain, June 27–30, 2017. Cham: Springer. CRM Ser. Math. Phys., 37-68 (2019).
Summary: 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.
For the entire collection see [Zbl 1421.81004].

MSC:
81Q60 Supersymmetry and quantum mechanics
81V70 Many-body theory; quantum Hall effect
81R30 Coherent states
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
81Q80 Special quantum systems, such as solvable systems
82D80 Statistical mechanical studies of nanostructures and nanoparticles
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