About Scattering on the Ring

Bogevolnov, V. B.; Mikhailova, A.; Павлов, Б. С.; Yafyasov, A. M.

doi:10.1007/978-3-0348-8323-8_8

Cited by 10 publications

(16 citation statements)

References 7 publications

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“…It appeared, that the singularities of the first and second term at the eigenvalues of L int compensate each other, so that only the zeros of the denominator DN −− + K − arise as singularities of DN Λ on ∆. A one-dimensional version of the statement can be found in [9] and a rescription of the classical Krein formula with compensated singularities is given in [44]. In this paper we review the compensation singularities in Theorem 3.1, following [2] for a general thin junction and prove a similar statement, see Theorem 3.3 for the Intermediate ND-map.…”

Section: Krein Formulae For the Intermediate Dn-map And Nd-map With mentioning

confidence: 99%

“…We denote the corresponding boundary currents as ∂ϕ s ∂n Γ =: J s , ∂ϕ a ∂n Γ =: J a and consider the projections of them P + J sym , P + J asym onto the entrance space of the first (open) channel. We assume that the temperature is low, so that the role of an essential spectral interval is played by ∆ T = [9,11]. Then the eigenfunctions and eigenvalues of the intermediate Hamiltonian can be found based on Theorem 3.1, taking into account the approximate calculation of the potential Q(λ) of L ∆ (λ) :…”

Section: Symmetric Junctionmentioning

confidence: 99%

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A Solvable Model for Scattering on a Junction and a Modified Analytic Perturbation Procedure

Павлов

2009

Characteristic Functions, Scattering Functions and Transfer Functions

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The paper is dedicated to Mihail Samoilovich Livshits, who was first to consider a nonselfadjoit operator as a part of an extended sefadjoint scattering system.Abstract. We consider a one-body spin-less electron spectral problem for a resonance scattering system constructed of a quantum well weakly connected to a noncompact exterior reservoir, where the electron is free. The simplest kind of the resonance scattering system is a quantum network, with the reservoir composed of few disjoint cylindrical quantum wires, and the Schrödinger equation on the network, with the real bounded potential on the wells and constant potential on the wires. We propose a Dirichlet-to-Neumann -based analysis to reveal the resonance nature of conductance across the star-shaped element of the network (a junction), derive an approximate formula for the scattering matrix of the junction, construct a fitted zero-range solvable model of the junction and interpret a phenomenological parameter arising in DattaDas Sarma boundary condition, see [13], for T-junctions. We also propose using of the fitted zero-range solvable model as the first step in a modified analytic perturbation procedure of calculation of the corresponding scattering matrix.

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Section: Krein Formulae For the Intermediate Dn-map And Nd-map With mentioning

confidence: 99%

Section: Symmetric Junctionmentioning

confidence: 99%

A Solvable Model for Scattering on a Junction and a Modified Analytic Perturbation Procedure

Павлов

2009

Characteristic Functions, Scattering Functions and Transfer Functions

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“…Hence, for small δ, only the poles at the zeros of the denominator of (5.3) give rise to the eigenvalues of the intermediate Hamiltonian. This statement, as a "lemma on compensation of singularities" was discovered [10] for a solvable model of network in form of a quantum graph, and later ( [59] and [58]) for quantum networks with nontrivial vertex domain. We prove it here, for the convenience of the reader, essentially following the version of the proof , suggested in [70].…”

Section: One-pole Approximation Of the Intermediate Dn-map For A Shrimentioning

confidence: 99%

“…We are able to prove a weaker version of this statement, for the networks which are "thin on the Fermi level in closed channels, for given temperature": see the definition and discussion in next paragraph. The one-dimensional variant of this statement was discovered [10] as a "Lemma on compensation of singularities", and it was used in [58] and [70] for analysis of spectral properties of resonances.…”

Section: Intermediate Dn-map Via Analytic Renormalization Proceduresmentioning

confidence: 99%

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Intermediate Hamiltonian via Glazman's splitting and analytic perturbation for meromorphic matrix‐functions

Mikhailova

Павлов

Prokhorov

2007

Mathematische Nachrichten

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We consider the spectral problem for the Schrödinger operator L on a quantum network, represented as a union of a finite number of quantum wells and finite or semi-infinite straight quantum wires connecting them to each other and to infinity. The absolutely continuous spectrum of the Schrödinger operator, with real piecewisecontinuous potential on the wells and constant potential on the wires, has a band structure defined by the geometry and the potentials on the semi-infinite wires. We split the Schrödinger operator, depending on the Fermi level Λ, into an orthogonal sum of two operators L → lΛ ⊕ LΛ, which have the complementary branches of the absolutely continuous spectra coincident, counting multiplicity, respectively with the unionsof absolutely continuous branches σ = [λ σ , ∞) of open and closed channels in the semi-infinite wires, with thresholds λσ < Λ, λσ > Λ, respectively. This enables us to reduce the problem of evaluation of the scattering matrix on the network to the solution of a finite linear system, involving the Dirichlet-to-Neumann map of the intermediate Hamiltonian LΛ. We calculate the intermediate Dirichlet-to-Neumann map approximately, for thin networks, based on the classical DN-map of the relevant Schrödinger operator on the compact part of the network. To this end we develop a modified analytic perturbation procedure for meromorphic operatorfunctions with small one-dimensional residues.

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