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Título
On supersymmetric Dirac delta interactions
Autor(es)
Materia
Supersymmetry
Physics
Supersymmetric models
Fecha de publicación
2014-01-28
Editor
Elsevier
Resumen
[EN] 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-
δ
s 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 eigen-
functions 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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- MATHPHYS. Artículos [94]