On the existence of resonances in the transmission probability for interactions arising from derivatives of Dirac s delta function

Christiansen, Peter Leth; Arnbak, H C; Zolotaryuk, A. V.; Ermakov, V. N.; Gaididei, Yuri

doi:10.1088/0305-4470/36/27/311

Cited by 84 publications

(189 citation statements)

References 27 publications

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“…[12]. In that paper, an exactly solvable model (1) with a specially chosen step function V = αΨ was considered.…”

Section: Introductionmentioning

confidence: 99%

On norm resolvent convergence of Schrödinger operators with δ′-like potentials

Golovaty¹,

Hryniv²

2010

J. Phys. A: Math. Theor.

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Abstract. For a function V : R → R that is integrable and compactly supported, we prove the norm resolvent convergence, as ε → 0, of a family S ε of one-dimensional Schrödinger operators on the line of the formthen the functions ε −2 V (x/ε) converge in the sense of distributions as ε → 0 to δ ′ (x), and the limit S 0 of S ε might be considered as a 'physically motivated' interpretation of the one-dimensional Schrödinger operator with potential δ ′ . In 1985,Šeba claimed that the limit operator S 0 is the direct sum of the free Schrödinger operators on positive and negative semi-axes subject to the Dirichlet condition at x = 0, which suggested that in dimension 1 there is no non-trivial Hamiltonian with potential δ ′ . In this paper, we show that in fact S 0 essentially depends on V : although the above results are true generically, in the exceptional (or 'resonant') case, the limit S 0 is non-trivial and is determined by the properties of an auxiliary Sturm-Liouville spectral problem associated with V . We then set V (ξ) = αΨ(ξ) with a fixed Ψ and show that there exists a countable set of resonances {α k } ∞ k=−∞ for which a partial transmission of the wave package occurs for S 0 .

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“…[12]. In that paper, an exactly solvable model (1) with a specially chosen step function V = αΨ was considered.…”

Section: Introductionmentioning

confidence: 99%

On norm resolvent convergence of Schrödinger operators with δ′-like potentials

Golovaty¹,

Hryniv²

2010

J. Phys. A: Math. Theor.

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“…Type II can be found only if the potential is asymmetric as a function of x [16]. The partial transparency that was found in [13] is an illustration of the threshold anomaly of type II. Let us add that threshold anomaly does not occur to a potential of the form of ∆ (x) of Eq.…”

Section: The Derivative Of the Delta Functionmentioning

confidence: 90%

“…This is in the following sense. Recently Christiansen et al [13] re-examined the transmission-reflection problem with a potential of the form of δ (x). They assumed the δ (x) as the narrow width limit of the following rectangular function [see their Eq.…”

Section: The Derivative Of the Delta Functionmentioning

confidence: 99%

“…In the Schrödinger equation with V σ (x), if we introduce dimensionless quantities y = x/ and η = k, we can eliminate x and k in favor of y and η. The transmission coefficient can be expressed in terms of dimensionless parameters σ and η [13]. In the limit of → 0 and η → 0, the transmission coefficient is reduced to a function of σ alone, which is independent of k; see Eq.…”

Section: The Derivative Of the Delta Functionmentioning

confidence: 99%

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Unusual situations that arise with the Dirac delta function and its derivative

Coutinho

Nogami

Toyama

2009

Rev. Bras. Ensino Fís.

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There is a situation such that, when a function ƒ(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">) is combined with the Dirac delta function δ(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">), the usual formula <img src="/img/revistas/rbef/v31n4/a04form01.gif" align="absmiddle">does not hold. A similar situation may also be encountered with the derivative of the delta function δ'(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">), regarding the validity of <img src="/img/revistas/rbef/v31n4/a04form02.gif" align="absmiddle">. We present an overview of such unusual situations and elucidate their underlying mechanisms. We discuss implications of the situations regarding the transmission-reflection problem of one-dimensional quantum mechanics.

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“…in the norm resolvent sense) of sequences of operators of the form (1.2) H n = H 0 + B n , B n = B n · where B n belongs to some space of regular functions and satisfies B n −→ B in D ′ . Sequences of this kind have been used in many applications, (see, for instance, [30] and the references therein), and their convergence properties were studied for particular cases [1,15,16,18,30,44,47,48]. Unfortunately, the relation between the convergence of B n in D ′ and the convergence of the associated operators H n in the norm resolvent is not straightforward [12,28,29,48,49].…”

mentioning

confidence: 99%

One-dimensional Schrödinger operators with singular potentials: A Schwartz distributional formulation

Dias

Jorge

Prata

2016

Journal of Differential Equations

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Abstract. Using an extension of the Hörmander product of distributions, we obtain an intrinsic formulation of one-dimensional Schrödinger operators with singular potentials. This formulation is entirely defined in terms of standard Schwartz distributions and does not require (as some previous approaches) the use of more general distributions or generalized functions. We determine, in the new formulation, the action and domain of the Schrödinger operators with arbitrary singular boundary potentials. We also consider the inverse problem, and obtain a general procedure for constructing the singular (pseudo) potential that imposes a specific set of (local) boundary conditions. This procedure is used to determine the boundary operators for the complete four-parameter family of one-dimensional Schrödinger operators with a point interaction. Finally, the δ and δ ′ potentials are studied in detail, and the corresponding Schrödinger operators are shown to coincide with the norm resolvent limit of specific sequences of Schrödinger operators with regular potentials.

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On the existence of resonances in the transmission probability for interactions arising from derivatives of Dirac s delta function

Cited by 84 publications

References 27 publications

On norm resolvent convergence of Schrödinger operators with δ′-like potentials

On norm resolvent convergence of Schrödinger operators with δ′-like potentials

Unusual situations that arise with the Dirac delta function and its derivative

One-dimensional Schrödinger operators with singular potentials: A Schwartz distributional formulation

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