One-Dimensional Tunnel-Junction Formula for the Schrödinger Particle

Hirokawa, Masao; Kosaka, Takuya

doi:10.1137/130929072

Cited by 2 publications

(4 citation statements)

References 8 publications

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“…Let us make small two remarks at the tail end of this paper. Using our method, we can completely classify the boundary conditions of all self-adjoint extensions of the minimal Schrödinger operator, too [19]. In the Dirac operator's case, there is no effect of the length of junction in the boundary condition.…”

Section: Resultsmentioning

confidence: 99%

“…For the case where the electron's wave functions do not pass through the junction, their boundary condition can be described by two parameters, γ L , γ Let us make small two remarks at the tail end of this paper. Using our method, we can completely classify the boundary conditions of all self-adjoint extensions of the minimal Schrödinger operator, too [19]. In the Dirac operator's case, there is no effect of the length of junction in the boundary condition.…”

Section: Discussionmentioning

confidence: 99%

“…To obtain our formulae, we invent our representation of U(2) in Proposition 4.3. We then realize that all the boundary conditions are independent of 2Λ, the length of the junction, which is different from the case for the Schrödinger operator [15,19,31]. Through our mathematical toy model we propose a mathematically fundamental idea for a tunnel-junction device in the light of the industry of quantum engineering.…”

Section: Introductionmentioning

confidence: 97%

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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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Section: Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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A mathematical aspect of a tunnel-junction for spintronic qubit

Hirokawa

Kosaka

2014

Journal of Mathematical Analysis and Applications

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“…These objects yield exactly solvable models [2,4,5,6,8,11,12,26,38,45] and have been widely used in applications in quantum mechanics (e.g. in models of low-energy scattering [3,13,14,35] and quantum systems with boundaries [22,23,24,27,32]), condensed matter physics [10,17,25] and, more recently, on the approximation of thin quantum waveguides by quantum graphs [1,15,16,20].…”

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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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One-Dimensional Tunnel-Junction Formula for the Schrödinger Particle

Cited by 2 publications

References 8 publications

A mathematical aspect of a tunnel-junction for spintronic qubit

A mathematical aspect of a tunnel-junction for spintronic qubit

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

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