Boundary conditions for the states with resonant tunnelling across the -potential

Zolotaryuk, A. V.

doi:10.1016/j.physleta.2010.02.005

Cited by 48 publications

(86 citation statements)

References 26 publications

(86 reference statements)

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“…The latter feature is in perfect agreement with the well-known findings about the Hamiltonian [26][27][28][29][30][31] (the reader also interested in the literature on the potential related to the derivative of the Dirac distribution is referred to [26,32,33]), where the coupling constant β of the interaction also enters in the condition…”

Section: The Rigorous Definition Of H βsupporting

confidence: 87%

A remarkable spectral feature of the Schrödinger Hamiltonian of the harmonic oscillator perturbed by an attractive δ′-interaction centred at the origin: double degeneracy and level crossing

Albeverio¹,

Fassari²,

Rinaldi³

2013

J. Phys. A: Math. Theor.

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We rigorously define the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive δ -interaction, of strength β, centred at 0 (the bottom of the confining parabolic potential), by explicitly providing its resolvent. Our approach is based on a 'coupling constant renormalization', related to a technique originated in quantum field theory and implemented in the rigorous mathematical construction of the self-adjoint operator representing the negative Laplacian perturbed by the δ-interaction in two and three dimensions. The way the δ -interaction enters in our Hamiltonian corresponds to the one originally discussed for the free Hamiltonian (instead of the harmonic oscillator one) by P Sěba. It should not be confused with the δ -potential perturbation of the harmonic oscillator discussed, e.g., in a recent paper by Gadella, Glasser and Nieto (also introduced by P Sěba as a perturbation of the one-dimensional free Laplacian and recently investigated in that context by Golovaty, Hryniv and Zolotaryuk). We investigate in detail the spectrum of our perturbed harmonic oscillator. The spectral structure differs from that of the one-dimensional harmonic oscillator perturbed by an attractive δ-interaction centred at the origin: the even eigenvalues are not modified at all by the δ -interaction. Moreover, all the odd eigenvalues, regarded as functions of β, exhibit the rather remarkable phenomenon called 'level crossing' after first producing the double degeneracy of all the even eigenvalues for the value β = β 0 = being the beta function).

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Section: The Rigorous Definition Of H βsupporting

confidence: 87%

A remarkable spectral feature of the Schrödinger Hamiltonian of the harmonic oscillator perturbed by an attractive δ′-interaction centred at the origin: double degeneracy and level crossing

Albeverio¹,

Fassari²,

Rinaldi³

2013

J. Phys. A: Math. Theor.

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“…However, the solution obtained by regularization is not unique and has been studied in more detail by Zolotaryuk et al [3] and Zolotaryuk [4]. Scattering potentials with a higher degree of singularity than the d potential were considered by Exner & Kondej [5], Zolotaryuk et al [6] and Zolotaryuk [7].…”

Section: Introductionmentioning

confidence: 99%

Non-relativistic and relativistic scattering by short-range potentials

Arnbak

Christiansen

Gaididei

2011

Phil. Trans. R. Soc. A.

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Relativistic and non-relativistic scattering by short-range potentials is investigated for selected problems. Scattering by the d potential in the Schrödinger equation and d potentials in the Dirac equation must be solved by regularization, efficiently carried out by a perturbation technique involving a stretched variable. Asymmetric regularizations yield non-unique scattering coefficients. Resonant penetration through the potentials is found. Approximative Schrödinger equations in the non-relativistic limit are discussed in detail.

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“…There have also been controversies in the literature due to conflicting regularizationdependent results obtained in the treatment of the so-called δ ′ -interaction [30][31][32][33]. These controversies, however, were clarified by using mathematically rigorous methods such as selfadjoint extensions (SAE) [34][35][36][37] and Schwartz's distribution theory [38,39] (there are also alternative approaches using non-standard distribution theory, e.g., [40,41]), both of which unequivocally demonstrate the existence of a four-parameter family of point interactions in one dimensional quantum mechanics-these include, but are not restricted to, the δ interaction as well as other interactions commonly associated with the δ ′ in the literature.…”

Section: Introductionmentioning

confidence: 99%

Double General Point Interactions: Symmetry and Tunneling Times

et al. 2016

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We consider the one dimensional problem of a non-relativistic quantum particle scattering off a double barrier built from two generalized point interactions (each one characterized as a member of the four parameter family of point interactions). The properties of the double point barrier under parity transformations are investigated, using the distributional approach, and the constraints on the parameters necessary for the interaction to have a well-defined parity are obtained. We show that the limit of zero interbarrier distance of a renormalized odd arrangement with two δ ′ is either trivial or does not exist as a generalized point interaction. Finally, we specialize to double barriers with defined parity, calculate the phase and Salecker-Wigner-Peres clock times and argue that the emergence of the generalized Hartman effect is an artifact of the extreme opaque limit.

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Boundary conditions for the states with resonant tunnelling across the -potential

Cited by 48 publications

References 26 publications

A remarkable spectral feature of the Schrödinger Hamiltonian of the harmonic oscillator perturbed by an attractive δ′-interaction centred at the origin: double degeneracy and level crossing

A remarkable spectral feature of the Schrödinger Hamiltonian of the harmonic oscillator perturbed by an attractive δ′-interaction centred at the origin: double degeneracy and level crossing

Non-relativistic and relativistic scattering by short-range potentials

Double General Point Interactions: Symmetry and Tunneling Times

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