Scattering on a Compact Domain with Few Semi-Infinite Wires Attached: Resonance Case

Mikhailova, A.; Павлов, Б. С.; Popov, I. Yu.; Rudakova, T. V.; Yafyasov, A. M.

doi:10.1002/1522-2616(200202)235:1<101::aid-mana101>3.0.co;2-v

Cited by 22 publications

(15 citation statements)

References 8 publications

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“…We shall not discuss this topic in the present paper. Important facts on the scattering solutions in networks of thin fibers can be found in [23], [22].…”

Section: Historical Remarksmentioning

confidence: 99%

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Laplace operator in networks of thin fibers: Spectrum near the threshold

Molchanov¹,

Vainberg²

2007

Stochastic Analysis in Mathematical Physics

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Our talk at Lisbon SAMP conference was based mainly on our recent results on small diameter asymptotics for solutions of the Helmgoltz equation in networks of thin fibers. These results were published in [21]. The present paper contains a detailed review of [21] under some assumptions which make the results much more transparent. It also contains several new theorems on the structure of the spectrum near the threshold. small diameter asymptotics of the resolvent, and solutions of the evolution equation.

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“…We shall not discuss this topic in the present paper. Important facts on the scattering solutions in networks of thin fibers can be found in [23], [22].…”

Section: Historical Remarksmentioning

confidence: 99%

“…For example, if Ω ε is a finite cylinder with the Dirichlet boundary condition (see the introduction) these eigenvalues have the form λ 0 + ε 2 m 2 /l 2 , m ≥ 1. Thus, assumption (22) allows one to study any finite number of eigenvalues near λ = λ 0 .…”

Section: Lemmamentioning

confidence: 99%

Laplace operator in networks of thin fibers: Spectrum near the threshold

Molchanov¹,

Vainberg²

2007

Stochastic Analysis in Mathematical Physics

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“…Sobolev & Solomyak (2002) have studied the effect of introducing a real perturbation of the zero potential on such an infinite homogeneous tree when the coupling constant tends to infinity. A further impetus has been given to these studies by the requirements in micro electronics fabrication problems, in particular the construction of quantum switches and other nano-computational devices (see Mikhailova et al 2002;Mikhaylova & Pavlov 2002;Pavlov 2002) where scattering problems on trees have been studied. In a recent paper, Pivovarchik (2000) considers the inverse spectral problem of the recovery of the coefficients of the Sturm-Liouville problem on a domain consisting of three intervals together with appropriate interface and boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

A Borg–Levinson theorem for trees

2005

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“…and conditions (22)- (24). Here ξ is a point of Γ which is not a vertex, and δ ξ (γ) is the delta function supported on γ = ξ.…”

Section: Introductionmentioning

confidence: 99%

Propagation of Waves in Networks of Thin Fibers

Molchanov¹,

Vainberg²

2009

Integral Methods in Science and Engineering, Volume 1

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The paper contains a simplified and improved version of the results obtained by the authors earlier. Wave propagation is discussed in a network of branched thin wave guides when the thickness vanishes and the wave guides shrink to a one dimensional graph. It is shown that asymptotically one can describe the propagating waves, the spectrum and the resolvent in terms of solutions of ordinary differential equations on the limiting graph. The vertices of the graph correspond to junctions of the wave guides. In order to determine the solutions of the ODE on the graph uniquely, one needs to know the gluing conditions (GC) on the vertices of the graph.Unlike other publications on this topic, we consider the situation when the spectral parameter is greater than the threshold, i.e., the propagation of waves is possible in cylindrical parts of the network. We show that the GC in this case can be expressed in terms of the scattering matrices related to individual junctions. The results are extended to the values of the spectral parameter below the threshold and around it. * The authors were supported partially by the NSF grant DMS-0706928.i.e. Ω ε is the ε-contraction of Ω = Ω 1 .For the sake of simplicity, we shall assume that Ω ε is self-similar in a neighborhood of each junction. Namely, let J j(v),ε be the junction which corresponds to a vertex v ∈ V of the limiting graph Γ. Consider a junction J v,ε = J j(v),ε and all the channels adjacent to J v,ε . If some of these channels have finite length, we extend them to infinity. We assume that, for each v ∈ V, the resulting domain Ω v,ε , which consists of the junction J v,ε and the semi-infinite channels emanating from it, is a spider domain. We also assume here that all the channels C j,ε have the same cross-section ω ε . This assumption is needed only to make the results more transparent (The general case is studied in [22]). From the self-similarity assumption it follows that ω ε is an ε−homothety of a bounded domain ω ⊂ R d−1 .Let λ 0 < λ 1 ≤ λ 2 ... be eigenvalues of the negative Laplacian −∆ d−1 in ω with the BC B 0 u = 0 on ∂ω where B 0 coincides with the boundary operator B on the channels, see (1), with ε = 1 in the case of the third boundary condition. Let {ϕ n (y)}, y ∈ ω ∈ R d−1 , be the set of corresponding orthonormal eigenfunctions. Then λ n are eigenvalues of −ε 2 ∆ d−1 in ω ε and {ε −d/2 ϕ n (y/ε)} are the corresponding eigenfunctions. In the presence of infinite channels, the spectrum of the operator H ε consists of an absolutely continuous component which coincides with the semi-bounded interval [λ 0 , ∞) and a discrete set of eigenvalues. The eigenvalues can be located below λ 0 and can be embedded into the absolutely continuous spectrum. We will call the point λ = λ 0 the threshold since it is the bottom of the absolutely continuous spectrum or (and) the first point of accumulation of the eigenvalues as ε → 0. Let us consider two of the simplest examples: the Dirichlet problem in a half infinite cylinder and in a bounded cylinder of length l. In the fir...

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Scattering on a Compact Domain with Few Semi-Infinite Wires Attached: Resonance Case

Cited by 22 publications

References 8 publications

Laplace operator in networks of thin fibers: Spectrum near the threshold

Laplace operator in networks of thin fibers: Spectrum near the threshold

A Borg–Levinson theorem for trees

Propagation of Waves in Networks of Thin Fibers

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