Mathematical Physics of Quantum Wires and Devices

Hurt, Norman E.

doi:10.1007/978-94-015-9626-8

Cited by 47 publications

(31 citation statements)

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“…The appearance of trapped modes (finite energy solutions) under perturbations has attracted a lot of research during the last decades (see [4,10]). Frequently, their existence is a consequence of the structure of the continuous spectrum of the unperturbed problem: when there is a cut-off of the continuous spectrum, then, under certain perturbations, an eigenfunction occurs.…”

Section: Introductionmentioning

confidence: 99%

Water waves trapped by thin submerged cylinders: Exact solutions

Garibay

Zhevandrov

2015

Russ. J. Math. Phys.

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Exact solutions of the linear water-wave problem describing oblique water waves trapped by a submerged horizontal cylinder of small (but otherwise arbitrary) cross-section are constructed in the form of a convergent series in powers of the small parameter characterizing the "thinness" of the cylinder. The terms of this series are expressed through the solutions of the exterior Neumann problem for the Laplace equation describing the flow of unbounded fluid past the cylinder.

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Section: Introductionmentioning

confidence: 99%

Water waves trapped by thin submerged cylinders: Exact solutions

Garibay

Zhevandrov

2015

Russ. J. Math. Phys.

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“…In such structures the mean free path can reach hundreds of micrometers, while the transverse confinement can be of the order of ten nanometers. Because of their small dimensions and purity such devices represent an ideal framework for the analysis of many peculiar phenomena of quantum mechanics, see, e.g., [18] and [30], and in many applications they can be treated as one-dimensional systems. For a comprehensive review on properties and applications of quantum graphs we refer to [4,[25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Graph-like models for thin waveguides with Robin boundary conditions

Cacciapuoti

Finco

2010

Asymptotic Analysis

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We discuss the limit of small width for the Laplacian defined on a waveguide with Robin boundary conditions. Under suitable hypothesis on the scaling of the curvature, we prove the convergence of the Robin Laplacian to the Laplacian on the corresponding graph. We show that in the limit of small width of the waveguide the transverse modes are independent. The projections on each transverse mode generically give rise to decoupling between the edges of the graph while exceptionally a coupling can occur. The coupling takes place if there exists a resonance at the threshold of the continuum spectrum of the effective Hamiltonian resulting from the projection.

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“…Remark 6.2. The functions Q and R seem completely different, but in fact they are related because g (1) and g (2) are related, see (4.15).…”

Section: Zeros Of the Reflection And Transmission Coefficients And Thmentioning

confidence: 99%

“…e.g., the books [1,2,3] and references therein). In our previous paper [4] we studied this phenomenon for the Rayleigh-Bloch waves generated by a weak periodic perturbation.…”

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confidence: 99%

Wave scattering by a periodic perturbation: embedded Rayleigh–Bloch modes and resonances

Zhevandrov

Merzon

Rodríguez

et al. 2019

Z. Angew. Math. Phys.

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The scattering of quasiperiodic waves for a two-dimensional Helmholtz equation with a constant refractive index perturbed by a function which is periodic in one direction and of finite support in the other is considered. The scattering problem is uniquely solvable for almost all frequencies and formulas of Breit-Wigner and Fano type for the reflection and transmission coefficients are obtained in a neighborhood of the resonance (a pole of the reflection coefficient). We indicate also the values of the parameters involved which provide total transmission and reflection. For some exceptional frequencies and perturbations (when the imaginary part of the resonance vanishes) the scattering problem is not uniquely solvable and in the latter case there exist embedded Rayleigh-Bloch modes whose frequencies are explicitly calculated in terms of infinite convergent series in powers of the small parameter characterizing the magnitude of the perturbation.

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Mathematical Physics of Quantum Wires and Devices

Cited by 47 publications

References 0 publications

Water waves trapped by thin submerged cylinders: Exact solutions

Water waves trapped by thin submerged cylinders: Exact solutions

Graph-like models for thin waveguides with Robin boundary conditions

Wave scattering by a periodic perturbation: embedded Rayleigh–Bloch modes and resonances

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