On Eisenbud's and Wigner's R-matrix: A general approach

Behrndt, Jussi; Neidhardt, Hagen; Racec, E. R.; Racec, Paul N.; Wulf, Ulrich

doi:10.1016/j.jde.2008.02.004

Cited by 9 publications

(8 citation statements)

References 37 publications

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“…In the case of the one-dimensional system without spherical symmetry, it was recently proven mathematically rigorous that the R-matrix formalism allows for a proper expansion of the scattering matrix on the real energy axis [28]. In this section we present an extension of the R-matrix formalism for 2D scattering problem.…”

Section: R-matrix Formalism For Two Dimensionsmentioning

confidence: 99%

“…The classically forbidden spectrum contains the bound states or the localized states. The R-matrix formalism can provide also these states, as long as the boundary points ±d z are far enough from the quantum dot, so that the bound states fulfill the Neumann boundary condition (28). In such a way, the energies of the bound states are the negative Wigner-Eisenbud energies and the wave functions for the bound state z, r ).…”

Section: Conical Quantum Dot Inside a Cylindrical Nanowirementioning

confidence: 99%

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R-matrix Formalism for Electron Scattering in Two Dimensions with Applications to Nanostructures with Quantum Dots

Racec

Neidhardt

2010

Engineering Materials

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We investigate the scattering phenomena in two dimensions produced by a general finite-range nonseparable potential. This situation can appear either in a Cartesian geometry or in a heterostructure with cylindrical symmetry. Increasing the dimensionality of the scattering problem, new processes as the scattering between conducting channels and the scattering from conducting to evanescent channels are allowed. For certain values of the energy called resonance energy, the transmission through the scattering region changes dramatically in comparison with an onedimensional problem. If the potential has an attractive character, even the evanescent channels can be seen as dips of the total transmission. The multi-channel current scattering matrix is determined using its representation in terms of the R-matrix. The resonant transmission peaks are characterized quantitatively through the poles of the current scattering matrix. Detailed maps of the localization probability density sustain the physical interpretation of the resonances. Our formalism is applied to a quantum dot in a two-dimensional electron gas and to a conical quantum dot embedded inside a cylindrical nanowire.

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Section: R-matrix Formalism For Two Dimensionsmentioning

confidence: 99%

Section: Conical Quantum Dot Inside a Cylindrical Nanowirementioning

confidence: 99%

R-matrix Formalism for Electron Scattering in Two Dimensions with Applications to Nanostructures with Quantum Dots

Racec

Neidhardt

2010

Engineering Materials

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“…In the case of the one-dimensional system it was recently proven mathematically rigorous that the R-matrix formalism allows for a proper expansion of the scattering matrix on the real energy axis 18 . In this section we present an extension of the R-matrix formalism for 2D scattering problem with cylindrical symmetry.…”

Section: R-matrix Formalism For Cylindrical Geometrymentioning

confidence: 99%

“…This reduces the scattering problem to two dimensions: r and z directions. Its solution is found numerically using the R-matrix formalism, 10,11,12,13,14,15,16,17,18 ex-tended for cylindrical coordinates.…”

Section: Introductionmentioning

confidence: 99%

Evanescent channels and scattering in cylindrical nanowire heterostructures

Racec,

Neidhardt

2008

Preprint

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We investigate the scattering phenomena produced by a general finite-range nonseparable potential in a multi-channel two-probe cylindrical nanowire heterostructure. The multi-channel current scattering matrix is efficiently computed using the R-matrix formalism extended for cylindrical coordinates. Considering the contribution of the evanescent channels to the scattering matrix, we are able to put in evidence the specific dips in the tunneling coefficient in the case of an attractive potential. The cylindrical symmetry cancels the "selection rules" known for Cartesian coordinates. If the attractive potential is superposed over a non-uniform potential along the nanowire, then resonant transmission peaks appear. We can characterize them quantitatively through the poles of the current scattering matrix. Detailed maps of the localization probability density sustain the physical interpretation of the resonances (dips and peaks). Our formalism is applied to a variety of model systems like a quantum dot, a core/shell quantum ring or a double-barrier, embedded into the nanocylinder.

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“…The boundary conditions of the self-adjoint extensions of the momentum operators have been studied by mathematicians [18,26]. We note that there is a general theory in mathematics, called boundary triple, to handle the boundary condition, and the theory has still been developed [6,7,11,10,13]. In Refs.…”

mentioning

confidence: 99%

A mathematical aspect of a tunnel-junction for spintronic qubit

Hirokawa

Kosaka

2014

Journal of Mathematical Analysis and Applications

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We consider the Dirac particle living in the 1-dimensional configuration space with a junction for a spintronic qubit. We give concrete formulae explicitly showing the one-to-one correspondence between every self-adjoint extension of the minimal Dirac operator and the boundary condition of the wave functions of the Dirac particle. We then show that the boundary conditions are classified into two types: one of them is characterized by two parameters and the other is by three parameters. Then, we show that Benvegnù and Dabrowski's four-parameter family can actually be characterized by three parameters, concerned with the reflection, penetration, and phase factor.Tunnel-Junction Formulae for Spintronic Qubit 2 as a black box so that it has mathematical arbitrariness. We regard the wires as the union of the intervals, (−∞, −Λ) ∪ (Λ, ∞), Λ ≥ 0, for mathematical simplicity. Namely, the segment [−Λ, Λ] with length 2Λ plays a role of the junction. Many physicists have investigated the individual boundary condition of the wave functions of the Dirac particle for the corresponding self-adjoint extension of the Dirac operator in the case of the point interaction (i.e., Λ = 0) [1,2,8,14,30]. Meanwhile, the Dirac operator consists of the combination of the Dirac matrices and the momentum operator of electron. The boundary conditions of the self-adjoint extensions of the momentum operators have been studied by mathematicians [18,26]. We note that there is a general theory in mathematics, called boundary triple, to handle the boundary condition, and the theory has still been developed [6,7,11,10,13]. In Refs. [15,31] the appearance of a phase factor was proved for the Schrödinger particle under the same mathematical set-up as ours. For our Dirac particle, in the case where Λ = 0, Benvegnù and Dabrowski showed that a phase factor appears in their four-parameter family (see Eq.(15) of Ref. [8]). In addition, they showed how the boundary condition affects the spin. The description of boundary condition by the individual one-parameter families have been studied in Refs. [1,14,20,30] (also see Eqs. (17) and (18) of Ref.[8]). We will go ahead and make an in-depth research for the Dirac particle. Thus, we employ the minimal Dirac operator for the Hamiltonian. In this paper, we follow the machinery in Refs. [8,15,18,26,31] based on the von Neumann's theory [27,34], in which all the self-adjoint extensions of our minimal Dirac operator are parameterized by U ∈ U(2), where U(2) is unitary group of degree 2.We prove that all the boundary conditions of wave functions of our Dirac particle are completely classified into the two types (See Corollary 4.5). One of them is the type that states the wave functions do not pass through the junction, and described by two parameters, γ L , γ R ∈ C with |γ L | = |γ R | = 1, concerned with the reflections at −Λ and at +Λ, respectively. The other is the type that states the wave functions do pass through the junction, and described by Benvegnù and Dabrowski's four-parameter family (see Theorem 4.2, Theo...

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On Eisenbud's and Wigner's R-matrix: A general approach

Cited by 9 publications

References 37 publications

R-matrix Formalism for Electron Scattering in Two Dimensions with Applications to Nanostructures with Quantum Dots

R-matrix Formalism for Electron Scattering in Two Dimensions with Applications to Nanostructures with Quantum Dots

Evanescent channels and scattering in cylindrical nanowire heterostructures

A mathematical aspect of a tunnel-junction for spintronic qubit

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