On the essential spectrum of quantum waveguides

Rabinovich, Vladimir; Castillo-Pérez, Raúl; Urbano-Altamirano, Francisco

doi:10.1002/mma.2623

Cited by 14 publications

(10 citation statements)

References 18 publications

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“…A general theory of the limit operators method for bounded operators has been presented in the book . In the papers, this method was applied to an investigation of the essential spectra of electromagnetic Schrödinger and Dirac operators on

R^{n}

for wide classes of potentials, and in the papers for an investigation of the essential spectra of discrete operators on

Z^{n}

, on periodic combinatorial and quantum graphs, and quantum waveguides.…”

Section: Introductionmentioning

confidence: 99%

Schrödinger operators with interactions on unbounded hypersurfaces

Rabinovich¹

2018

Math Methods in App Sciences

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We consider Schrödinger operators H = −Δ + W + Ws on Rn with regular potentials W∈L∞false(Rnfalse) and singular potentials Ws∈D′false(Rnfalse) with supports on unbounded enough smooth hypersurfaces Γ. In particular, we consider singular potentials that are linear combinations of δ−functions on Γ and its normal derivatives. We consider extensions of H as symmetric operators in L2false(Rnfalse) with domain C0∞false(Rn⃥normalΓfalse) to self‐adjoint operators scriptH in L2false(Rnfalse). These extensions are realized as operators of transmission problems for −Δ + W in the space H2false(Rn⃥normalΓfalse) with some transmission conditions on Γ. Applying this approach, we obtain an effective description of essential spectra of the described Schrödinger operators.

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R^{n}

for wide classes of potentials, and in the papers for an investigation of the essential spectra of discrete operators on

Z^{n}

, on periodic combinatorial and quantum graphs, and quantum waveguides.…”

Section: Introductionmentioning

confidence: 99%

Schrödinger operators with interactions on unbounded hypersurfaces

Rabinovich¹

2018

Math Methods in App Sciences

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“…The Schrödinger operator

scriptH

with domain

D_{H} = ({}, φ \in H^{2} false(D \times double-struckR false) : φ |_{\partial D \times double-struckR} = 0)

is self‐adjoint in L 2 (Π), and

scriptH

has the essential spectrum

s p_{e s s} scriptH = false[μ_{1}^{2}, + \infty false)

. Moreover,

scriptH

can have a discrete spectrum located on the set

false[\inf_{D \times double-struckR} W false(x, z false), μ_{1} false)

, and if

\inf_{D \times false[0, H false]} W false(x, z false) \geq μ_{1},

the operator

scriptH

has the essential spectrum only (see, for instance, Rabinovich et al).…”

Section: Analytical Expression For the Transition Matricesmentioning

confidence: 99%

“…The problem of the electron propagation in the quantum waveguides (quantum wires in physical terminology) involves in the last 2 decades a lot of attention because of current and future applications in the modern electronic technology (see, for instance, Duclos and Exner and Exner and Kovařík and references cited there). Spectral properties of quantum waveguides have been considered in the paper, and the numerical analysis of the quantum scattering in 1‐dimensional quantum waveguides has been considered in Rabinovich and Urbano‐Altamirano by means of the spectral parameter power series (SPPS) method.…”

Section: Introductionmentioning

confidence: 99%

“…For the calculation of the Jost solutions associated with the transition matrices, we use the SPPS method. The SPPS method has been discovered by Kravchenko and has been successfully applied to different problems of Mathematical Physics (see, for instance, literature and references cited therein). Applying the SPPS method together with the Lagrange method for solutions of ordinary differential equations, we obtain effective algorithms for calculations of the transition matrices.…”

Section: Introductionmentioning

confidence: 99%

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Transition matrices for quantum waveguides with impurities

Rabinovich

Urbano-Altamirano

2018

Math Methods in App Sciences

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We consider the electron propagation in a cylindrical quantum waveguide Π ¼ D× R where D is a bounded domain in R 2 described by the Dirichlet problem for the Schrödinger operatorVðxÞ is the transversal confinement potential, and Wðx; zÞ is the impurity potential. We construct the left and right transition matrices and give an numerical algorithm for their calculations based on the spectral parameter power series method.

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“…As was shown in a number of recent publications the SPPS representation provides an efficient and accurate method for solving initial value, boundary value and spectral problems (see [7], [8], [16], [21], [23], [22], [24], [25], [31], [32], [33], [35], [47]). In this paper we demonstrate this fact in application to equation (1.1).…”

Section: Introductionmentioning

confidence: 99%

Spectral parameter power series for Sturm-Liouville equations with a potential polynomially dependent on the spectral parameter and Zakharov-Shabat systems

Kravchenko

Torba

Velasco-García

2015

Journal of Mathematical Physics

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Abstract. A spectral parameter power series (SPPS) representation for solutions of Sturm-Liouville equations of the formis obtained. It allows one to write a general solution of the equation as a power series in terms of the spectral parameter λ. The coefficients of the series are given in terms of recursive integrals involving a particular solution of the equation (pu 0 ) +qu0 = 0. The convenient form of the solution of ( * ) provides an efficient numerical method for solving corresponding initial value, boundary value and spectral problems. A special case of the Sturm-Liouville equation ( * ) arises in relation with the Zakharov-Shabat system. We derive an SPPS representation for its general solution and consider other applications as the one-dimensional Dirac system and the equation describing a damped string. Several numerical examples illustrate the efficiency and the accuracy of the numerical method based on the SPPS representations which besides its natural advantages like the simplicity in implementation and accuracy is applicable to the problems admitting complex coefficients, spectral parameter dependent boundary conditions and complex spectrum.

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On the essential spectrum of quantum waveguides

Cited by 14 publications

References 18 publications

Schrödinger operators with interactions on unbounded hypersurfaces

Schrödinger operators with interactions on unbounded hypersurfaces

Transition matrices for quantum waveguides with impurities

Spectral parameter power series for Sturm-Liouville equations with a potential polynomially dependent on the spectral parameter and Zakharov-Shabat systems

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