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

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Characteristic Functions, Scattering Functions and Transfer Functions

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 197))

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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 Datta- Das Sarma boundary condition, see [14], 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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  1. V.A. Fock Institute for Physics of St.-Petersburg University, Petrodvorets, 198905, Russia

    B. Pavlov

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  1. B. Pavlov
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  1. Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva, 84105, Israel

    Daniel Alpay  & Victor Vinnikov  & 

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The paper is dedicated to Mihail Samoilovich Livshits, who was first to consider a nonselfadjoint operator as a part of an extended selfadjoint scattering system.

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Pavlov, B. (2009). A Solvable Model for Scattering on a Junction and a Modified Analytic Perturbation Procedure. In: Alpay, D., Vinnikov, V. (eds) Characteristic Functions, Scattering Functions and Transfer Functions. Operator Theory: Advances and Applications, vol 197. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0183-2_11

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