The energy level structure of a variety of one-dimensional confining potentials and the effects of a local singular perturbation

Glasser, M. L.; Nieto, L. M.

doi:10.1139/cjp-2015-0356

Cited by 24 publications

(33 citation statements)

References 24 publications

(35 reference statements)

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“…By setting: 9) and 10) it is rather straightforward to obtain the new limit of the resolvent of (2.10) once the ultraviolet cut-off gets removed, namely:…”

Section: The Rigorous Definition Of the Three-dimensional Hamiltonianmentioning

confidence: 99%

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The behaviour of the three-dimensional Hamiltonian -Δ+λ[δ(x+x0)+δ(x-x0)] as the distance between the two centres vanishes

Albeverio¹,

Fassari²,

Rinaldi³

2017

Nanosystems: Phys. Chem. Math.

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“…By setting: 9) and 10) it is rather straightforward to obtain the new limit of the resolvent of (2.10) once the ultraviolet cut-off gets removed, namely:…”

Section: The Rigorous Definition Of the Three-dimensional Hamiltonianmentioning

confidence: 99%

“…even though the resolvent of this three-dimensional operator will have to be determined first since, differently from its one-dimensional counterpart (see [10,11]), it is not yet explicitly known.…”

Section: Final Remarksmentioning

confidence: 99%

The behaviour of the three-dimensional Hamiltonian -Δ+λ[δ(x+x0)+δ(x-x0)] as the distance between the two centres vanishes

Albeverio¹,

Fassari²,

Rinaldi³

2017

Nanosystems: Phys. Chem. Math.

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“…Using the above input and (3.8) in [68], we can obtain its Green function (integral kernel of the resolvent), which is…”

Section: The Spectrum Of H 0 Perturbed By An Attractive δ-Interactionmentioning

confidence: 99%

Spectroscopy of a one-dimensional V-shaped quantum well with a point impurity

et al. 2018

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a b s t r a c tWe consider the one-dimensional Hamiltonian with a V-shaped, decorated with a point impurity of either δ-type, or local δ ′ -type or even nonlocal δ ′ -type, thus yielding three exactly solvable models. We analyse the behaviour of the change in the energy levels when an interaction of the type −λ δ(x) or −λ δ(x − x 0 ) is switched on. In the first case, even energy levels, pertaining to antisymmetric bound states, remain invariant with respect to λ even though odd energy levels, pertaining to symmetric bound states, decrease as λ increases. In the second, all energy levels decrease when the factor λ increases. A similar study has been performed for the so-called nonlocal δ ′ interaction, requiring a coupling constant renormalisation, which implies the replacement of the form factor λ by a renormalised form factor β. In terms of β, odd energy levels are unchanged. However, we show the existence of level crossings: after a fixed value of β the energy of each even level, with the natural exception of the first one, becomes lower than the constant energy of the previous odd level. Finally, we consider an interaction of the type −λδ(x)+µδ ′ (x), and analyse in detail the discrete spectrum of the resulting selfadjoint Hamiltonian.

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“…where b x,y,z are constants. Eigenstates and eigenenergies are well known and can be found in literature [70]. By using linear photon polarizations (without restriction of generality), from Eq.…”

Section: B Symmetric Linear Potentialmentioning

confidence: 99%

Two-photon processes based on quantum commutators

et al. 2018

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We developed a new method to calculate two-photon processes in quantum mechanics that replaces the infinite summation over the intermediate states by a perturbation expansion. This latter consists of a series of commutators that involve position, momentum and hamiltonian quantum operators. We analyzed several single-and many-particle cases for which a closed form solution to the perturbation expansion exists, as well as more complicated cases for which a solution is found by convergence. Throughout the article, Rayleigh and Raman scattering are taken as examples of two-photon processes. The present method provides a clear distinction between the Thomson scattering, regarded as classical scattering, and quantum contributions. Such a distinction let us derive general results concerning light scattering. Finally, possible extensions to the developed formalism are discussed.

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The energy level structure of a variety of one-dimensional confining potentials and the effects of a local singular perturbation

Cited by 24 publications

References 24 publications

The behaviour of the three-dimensional Hamiltonian -Δ+λ[δ(x+x0)+δ(x-x0)] as the distance between the two centres vanishes

The behaviour of the three-dimensional Hamiltonian -Δ+λ[δ(x+x0)+δ(x-x0)] as the distance between the two centres vanishes

Spectroscopy of a one-dimensional V-shaped quantum well with a point impurity

Two-photon processes based on quantum commutators

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