On the Supersymmetric Spectra of two Planar Integrable Quantum Systems

Leon, M. Gonzalez

doi:10.1090/conm/563/11165

Cited by 2 publications

(1 citation statement)

References 47 publications

(84 reference statements)

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“…The N -body Yang-Lieb-Liniger systems have N degrees of freedom and one might think in more than one zero energy ground state in the SUSY version of these systems. In fact, there are two zero energy ground states in the supersymmetric extension of the Euler-Coulomb problem, a charged particle moving on a plane under the action of two Coulombian centres of force, build in Reference [41].…”

Section: Discussionmentioning

confidence: 99%

On supersymmetric Dirac-delta interactions

2015

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In this paper we construct N = 2 supersymmetric (SUSY) quantum mechanics over several configurations of Dirac-δ potentials from one single delta to a Dirac "comb ". We show in detail how the building of supersymmetry on potentials with delta interactions placed in two or more points on the real line requires the inclusion of quasi-square wells. Therefore, the basic ingredient of a supersymmetric Hamiltonian containing two or more Dirac-δs is the singular potential formed by a Dirac-δ plus a step (θ) at the same point. In this δ/θ SUSY Hamiltonian there is only one singlet ground state of zero energy annihilated by the two supercharges or a doublet of ground states paired by supersymmetry of positive energy depending on the relation between the Dirac well strength and the height of the step potential. We find a scenario of either unbroken supersymmetry with Witten index one or supersymmetry breaking when there is one "bosonic"and one "fermionic"ground state such that the Witten index is zero. We explain next the different structure of the scattering waves produced by three δ/θ potentials with respect to the eigenfunctions arising in the non-SUSY case. In particular, many more bound states paired by supersymmetry exist within the supersymmetric framework compared with the non-SUSY problem. An infinite array of equally spaced δ-interactions of the same strength but alternatively attractive and repulsive are susceptible of being promoted to a N = 2 supersymmetric system. The Bloch's theorem for wave functions in periodic potentials prompts a band spectrum also paired by supersymmetry. Self-isospectrality between the two partner Hamiltonians is thus found. Zero energy ground states are the non-propagating band lower edges, which exist in the spectra of both the two diagonal operators forming the SUSY Hamiltonian. We find that for the SUSY Dirac "comb"the naif Witten index is zero but supersymmetry is unbroken.

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Section: Discussionmentioning

confidence: 99%

On supersymmetric Dirac-delta interactions

2015

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Elementary solutions of the quantum planar two-center problem

León

Guilarte

Mayado

2016

EPL

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The quantum problem of an electron moving in a plane under the field created by two Coulombian centers admits simple analytical solutions for some particular inter-center distances. These elementary eigenfunctions, akin to those found by Demkov for the analogous three dimensional problem, are calculated using the framework of quasi-exact solvability of a pair of entangled ODE's descendants from the Heun equation. A different but interesting situation arises when the two centers have the same strength. In this case completely elementary solutions do not exist.

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On the Supersymmetric Spectra of two Planar Integrable Quantum Systems

Cited by 2 publications

References 47 publications

On supersymmetric Dirac-delta interactions

On supersymmetric Dirac-delta interactions

Elementary solutions of the quantum planar two-center problem

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