Trends in supersymmetric quantum mechanics

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.

33 pages, 2 figures, submitted as a contribution to the monographic volume "Integrability, Supersymmetry and Coherent States", a volume in honour of Professor Véronique Hussin, small changes done, references added

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Cited by in corpus (13)

• Dirac electron in graphene with magnetic fields arising from first-order intertwining operators
• Electron in bilayer graphene with magnetic fields leading to shape invariant potentials
• Bilayer graphene in magnetic fields generated by supersymmetry
• New solutions for graphene with scalar potentials by means of generalized intertwining
• Confluent Second-Order Supersymmetric Quantum Mechanics and Spectral Design
• Supersymmetrization of the Franke-Gorini-Kossakowski-Lindblad-Sudarshan equation
• Phase-space representation of coherent states generated through SUSY QM for tilted anisotropic Dirac materials
• Extended supersymmetry with central charges in higher dimensional Dirac action
• Probing Non-perturbative Supersymmetry Breaking through Lattice Path Integrals
• Supersymmetric Quantum Mechanics, multiphoton algebras and coherent states
• Exactly Solvable Sextic Potential Having Symmetric Triple-Well Structure
• Solvable Schrodinger Equations of Shape Invariant Potentials with Superpotential $W(x,A,B)=A\tanh 3px-B\coth px$
• Solvable Schrodinger Equations of Shape Invariant Potentials Having Superpotential W(x,A,B)=Atanh(px)+Btanh(6px)