A Study of Resonances in a One-Dimensional Model with Singular Hamiltonian and Mass Jumps

Alvarez, Juan Jose; Gadella, M.; Nieto, L. M.

doi:10.1007/s10773-010-0651-4

Cited by 12 publications

(12 citation statements)

References 28 publications

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“…where the singular (contact) terms are already included. A comment on the δ contribution in (14) is in order here. As explained in "Appendix Appendix A", the δ − δ perturbation is defined using the formalism of self-adjoint extensions of symmetric (formally Hermitian) operators with equal deficiency indices.…”

Section: Model and Motivationmentioning

confidence: 99%

“…Thus, bearing in mind that R a, we can consider the above simplified one-dimensional Hamiltonian (14) as a mean-field potential to describe neutron energy levels. One of the main advantages is that the eigenvalue equation H sing u (r ) = E n j u (r ) can be solved exactly for the wave function 1 in terms of Bessel functions.…”

Section: Model and Motivationmentioning

confidence: 99%

“…(15). These requirements are given by matching conditions relating the function u (r ) and its first derivative at the limit values of R. They can be written in terms of a SL(2, R) matrix as [14,15,50,51]…”

Section: Matching Conditionsmentioning

confidence: 99%

“…From a purely mathematical point of view, one-dimen-sion-al contact potentials have been studied as self-adjoint extensions of the kinetic energy operator −d 2 /dx 2 [3,12,13]. This approach has been used to construct several one-dimensional models which go beyond the Dirac delta potential [14,15]. A discussion on the physical meaning of the one-dimensional contact potentials constructed as self-adjoint extensions of the kinetic energy operator is given in [16,17].…”

Section: Introductionmentioning

confidence: 99%

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An approximation to the Woods–Saxon potential based on a contact interaction

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We study a non-relativistic particle subject to a three-dimensional spherical potential consisting of a finite well and a radial δ − δ contact interaction at the well edge. This contact potential is defined by appropriate matching conditions for the radial functions, thereby fixing a self-adjoint extension of the non-singular Hamiltonian. Since this model admits exact solutions for the wave function, we are able to characterize and calculate the number of bound states. We also extend some well-known properties of certain spherically symmetric potentials and describe the resonances, defined as unstable quantum states. Based on the Woods-Saxon potential, this configuration is implemented as a first approximation for a mean-field nuclear model. The results derived are tested with experimental and numerical data in the double magic nuclei 132 Sn and 208 Pb with an extra neutron.

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Section: Model and Motivationmentioning

confidence: 99%

Section: Model and Motivationmentioning

confidence: 99%

Section: Matching Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An approximation to the Woods–Saxon potential based on a contact interaction

et al. 2020

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“…The effective mass is then an operator which does not commute with momentum implying that the usual kinetic operator p 2 /2m(x) is not Hermitian and have to be modified. Effective mass operators have been largely used to obtain the minibands associated with periodicity of heterostructures [5,6,7,8,9,10,11].…”

mentioning

confidence: 99%

Indirect band gap in graphene from modulation of the Fermi velocity

Lima

Moraes

2015

Solid State Communications

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In this work we study theoretically the electronic properties of a sheet of graphene grown on a periodic heterostructure substrate. We write an effective Dirac equation, which includes a dependence of both the band gap and the Fermi velocity on the position, due to the influence of the substrate. This way, both bandgap and Fermi velocity enter the Dirac equation as operators. The Dirac equation is solved exactly and we find the superlattice minibands with gaps due to the breaking of translational symmetry induced by the underlying heterostructure. The spatial dependence of the Fermi velocity makes the band gap be indirect, bringing about interesting possibilities for applications in the design of nanoelectronic devices. In the limit of constant Fermi velocity we obtain a band structure, with direct band gap, very close to the one previously found in the literature, obtained using the transfer matrix method

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The Lippmann–Schwinger Formula and One Dimensional Models with Dirac Delta Interactions

Erman

Gadella

Uncu

2019

Integrability, Supersymmetry and Coherent States

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We show how a proper use of the Lippmann-Schwinger equation simplifies the calculations to obtain scattering states for one dimensional systems perturbed by N Dirac delta equations. Here, we consider two situations. In the former, attractive Dirac deltas perturbed the free one dimensional Schrödinger Hamiltonian. We obtain explicit expressions for scattering and Gamow states. For completeness, we show that the method to obtain bound states use comparable formulas, although not based on the Lippmann-Schwinger equation. Then, the attractive N deltas perturbed the one dimensional Salpeter equation. We also obtain explicit expressions for the scattering wave functions. Here, we need regularisation techniques that we implement via heat kernel regularisation.

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A Study of Resonances in a One-Dimensional Model with Singular Hamiltonian and Mass Jumps

Cited by 12 publications

References 28 publications

An approximation to the Woods–Saxon potential based on a contact interaction

An approximation to the Woods–Saxon potential based on a contact interaction

Indirect band gap in graphene from modulation of the Fermi velocity

The Lippmann–Schwinger Formula and One Dimensional Models with Dirac Delta Interactions

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