Non-relativistic and relativistic scattering by short-range potentials

Arnbak, H C; Christiansen, Peter Leth; Gaididei, Yuri

doi:10.1098/rsta.2010.0330

Cited by 15 publications

(22 citation statements)

References 15 publications

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“…Finally, for τ > 2 (at the line L 0 ) the divergent terms in (28) and (30) vanish at all at the same resonance conditions (47). Next, from expressions (27) and (29) we conclude that the limit Λ-matrix is of form (8) with the element θ defined by formulae (50). Similarly, this family of point interactions may be called 'resonant-tunnelling δ ′potentials of the J -type'.…”

Section: Splitting Of Resonance Sets At the Critical Point µ =mentioning

confidence: 84%

Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

Zolotaryuk¹

2017

J. Phys. A: Math. Theor.

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Abstract. Several families of one-point interactions are derived from the system consisting of two and three δ-potentials which are regularized by piecewise constant functions. In physical terms such an approximating system represents two or three extremely thin layers separated by some distance. The two-scale squeezing of this heterostructure to one point as both the width of δ-approximating functions and the distance between these functions simultaneously tend to zero is studied using the power parameterization through a squeezing parameter ε → 0, so that the intensity of each δ-potential is c j = a j ε 1−µ , a j ∈ R, j = 1, 2, 3, the width of each layer l = ε and the distance between the layers r = cε τ , c > 0. It is shown that at some values of intensities a 1 , a 2 and a 3 , the transmission across the limit point interactions is non-zero, whereas outside these (resonance) values the one-point interactions are opaque splitting the system at the point of singularity into two independent subsystems. Within the interval 1 < µ < 2, the resonance sets consist of two curves on the (a 1 , a 2 )-plane and three disconnected surfaces in the (a 1 , a 2 , a 3 )-space. While approaching the parameter µ to the critical value µ = 2, three types of splitting these sets into countable families of resonance curves and surfaces are observed.

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Section: Splitting Of Resonance Sets At the Critical Point µ =mentioning

confidence: 84%

Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

Zolotaryuk¹

2017

J. Phys. A: Math. Theor.

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“…Equation (9) is a direct consequence of requirements (i) and (ii) and the solution of the distributional Schrödinger equation with the general interaction (5). This alone, however, does not impose any constraints on the form of M ± (i.e., on the functionals α 0 and α 1 ); such constraints will come from imposing, in addition to (i) and (ii), condition (iii).…”

Section: Schrödinger's Equation With Point Interactionsmentioning

confidence: 99%

“…Point (zero-range) interactions have attracted great interest in quantum mechanics [1,2,3,4,5,6,7]. They provide important solvable models with a wide variety of applications in atomic physics, such as the Lieb-Liniger [8,9] model for a one dimensional gas of bosons interacting by means of a δ-function potential (for other applications see, e.g., [10,11,12] and references therein).…”

Section: Introductionmentioning

confidence: 99%

“…Given the somewhat abstract and mathematically involved nature of SAE, another approach was developed for this subject, consisting in the investigation of regularizations using sequences of regular short-range potentials which converge to point potentials in the zero-range limit [16,20,21,25]. Even if this method is more intuitive and appealing from a physicist's point of view, such an approach has often led to ambiguous and even contradictory results -which, in general, arise from the dependence of the particular regularizing scheme employed (see, for instance, [5,7,26] and references there cited).…”

Section: Introductionmentioning

confidence: 99%

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Distributional approach to point interactions in one-dimensional quantum mechanics

et al. 2014

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We consider the one-dimensional quantum mechanical problem of defining interactions concentrated at a single point in the framework of the theory of distributions. The often ill-defined product which describes the interaction term in the Schrödinger and Dirac equations is replaced by a well-defined distribution satisfying some simple mathematical conditions and, in addition, the physical requirement of probability current conservation is imposed. A four-parameter family of interactions thus emerges as the most general point interaction both in the non-relativistic and in the relativistic theories (in agreement with results obtained by self-adjoint extensions). Since the interaction is given explicitly, the distributional method allows one to carry out symmetry investigations in a simple way, and it proves to be useful to clarify some ambiguities related to the so-called δ interaction.

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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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Non-relativistic and relativistic scattering by short-range potentials

Cited by 15 publications

References 15 publications

Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

Distributional approach to point interactions in one-dimensional quantum mechanics

Double General Point Interactions: Symmetry and Tunneling Times

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