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30 pages, 1609 KiB

Open AccessArticle

Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian: Special Cases

by Manuel Gadella and Carlos San Millán

Symmetry 2022, 14(7), 1314; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14071314 - 24 Jun 2022

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In a previous paper, we used a classification of the self adjoint extensions, also called self-adjoint determinations, of the differential operator

- d^{2} / d x^{2}

in order to obtain the whole list of Supersymmetric (SUSY) partners of those selfadjoint determinations [...] Read more.

In a previous paper, we used a classification of the self adjoint extensions, also called self-adjoint determinations, of the differential operator

- d^{2} / d x^{2}

in order to obtain the whole list of Supersymmetric (SUSY) partners of those selfadjoint determinations for which the ground state has strictly positive energy. The existence of self adjoint determinations with a ground state of zero or even negative energy is a proved fact. In this paper, we analyze the possibility of constructing SUSY partners for those determinations. We also study those cases for which the ground state has a degeneracy, the study of their SUSY partners should be analyzed separately. So far, we have studied those determinations having an exactly solvable eigenvalue problem. On the present study, we also included some comments in relation to determinations not exactly solvable from this point of view. In addition, the use of self adjoint determinations for which the ground state wave function has nodes (zeroes) produces formal SUSY partners with a finite number of eigenvalues or even with a purely continuous spectrum. We give some worked examples of these situations. Full article

(This article belongs to the Special Issue Contact Interactions and Solvable Models in Quantum Mechanics)

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8 pages, 362 KiB

Open AccessArticle

Exact Treatment of the Ground States of Three Two-Dimensional Contact Interactions in a Uniform Magnetic Field

by Mervyn Lawrence Glasser

Symmetry 2022, 14(3), 489; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14030489 - 28 Feb 2022

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Abstract

The Green function for an electron confined to a plane in the presence of a uniform perpendicular magnetic field is derived and is decorated by several modified Dirac delta functions (MDF). In each case the ground state energy is examined as a function of the field strength, the spacing and the coupling constant of the MDFs. The results suggest that the magnetic field is ionising and the strength of the MDFs must attain a field-dependent critical value in order that a bound state be formed. This offers a new perspective on the question of whether a static magnetic field can be ionising. Full article

(This article belongs to the Special Issue Contact Interactions and Solvable Models in Quantum Mechanics)

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Other

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18 pages, 12902 KiB

Open AccessProject Report

Dyadic Helmholtz Green’s Function for Electromagnetic Wave Transmission/Diffraction through a Subwavelength Nano-Hole in a 2D Quantum Plasmonic Layer: An Exact Solution Using “Contact Potential”-like Dirac Delta Functions

by Désiré Miessein, Norman J. M. Horing and Harry Lenzing

Symmetry 2022, 14(6), 1134; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14061134 - 31 May 2022

Viewed by 1446

Abstract

The dyadic Helmholtz Green’s function for electromagnetic (EM) wave transmission/ diffraction through a subwavelength nano-hole in a two-dimensional (2D) plasmonic layer is discussed here analytically and numerically, employing “contact potential”-like Dirac delta functions in 1 and 2 dimensions (

δ (z)

[...] Read more.

δ (z)

and

δ (x) δ (y) \equiv δ^{(2)} (\vec{r})

). This analysis is carried out employing a succession of two coupled integral equations. The first integral equation determines the dyadic electromagnetic Green’s function

{\hat{G}}_{f s}

for the full non-perforated 2D quantum plasma layer in terms of the bulk 3D infinite-space dyadic electromagnetic Green’s function

{\hat{G}}_{3 D}

, with

δ (z)

representing the confinement of finite quantum plasma conductivity to the plane of the plasma layer at

z = 0

. The second integral equation determines the dyadic electromagnetic “hole” Green’s function

{\hat{G}}_{h o l e}

for the perforated 2D quantum plasma layer (containing the nano-hole) in terms of the dyadic electromagnetic Green’s function

{\hat{G}}_{f s}

for the full non-perforated 2D plasma layer, with

δ^{(2)} (\vec{r})

describing the exclusion of the quantum plasma layer conductivity properties from the nano-hole region in the vicinity of

\vec{r} = 0

on the plane. Taking the radius of the subwavelength nano-hole to be the smallest length scale of the system in conjunction with the 2D Dirac delta function representation of the excluded nano-hole plasma conductivity, both of the successive coupled integral equations are solved exactly, and we present a thorough numerical analysis (based on the exact analytic solution) for the resulting dyadic “hole” Green’s function

{\hat{G}}_{h o l e}

in full detail in both 3D and density plots. This result has been successfully applied to the determination of electromagnetic wave transmission/diffraction through the nano-hole of the perforated quantum plasmonic layer, jointly with the EM wave transmission through the rest of the plasma layer. This success necessarily involves spatial translational asymmetry induced by the use of spatial Dirac delta functions confining finite conductivity to the 2D quantum plasma sheet and the excision at a bit of it about the origin to represent the nano-hole perforation, thus breaking spatial translational invariance symmetry. Full article

(This article belongs to the Special Issue Contact Interactions and Solvable Models in Quantum Mechanics)

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