Waveguides with Combined Dirichlet and Robin Boundary Conditions

Freitas, Pedro; Krejčiřı́k, David

doi:10.1007/s11040-007-9015-6

Cited by 14 publications

(17 citation statements)

References 32 publications

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“…where the first inequality was established (for the unitarily equivalent operator H ε ) in the beginning of Section 2.3, the second inequality holds due to the monotonicity of ε → ν(ε) proved in [11], Theorem 2, and the equality follows from (2.7). (Alternatively, we could use Th.…”

Section: The Reference Hilbert Space and The Resultsmentioning

confidence: 94%

“…Let us denote the corresponding Laplacian by −Δ Ωε DRα . Then the method of the present paper gives Let us mention that strips with this combination of Dirichlet and Robin boundary conditions were studied in [11].…”

Section: Different Boundary Conditionsmentioning

confidence: 99%

“…κ < 0 in our setting), and that (b) holds. The results were further generalized in [11,21], numerically tested in [24], and established in a different physical model in [17].…”

Section: Introductionmentioning

confidence: 99%

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Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions

Krejčiřı́k¹

2008

ESAIM: COCV

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Abstract.We consider the Laplacian in a domain squeezed between two parallel curves in the plane, subject to Dirichlet boundary conditions on one of the curves and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the curves tends to zero. The asymptotics are uniform and local in the sense that the coefficients depend only on the extremal points where the ratio of the curvature radii of the Neumann boundary to the Dirichlet one is the biggest. We also show that the asymptotics can be obtained from a form of norm-resolvent convergence which takes into account the width-dependence of the domain of definition of the operators involved.Mathematics Subject Classification. 35P15, 49R50, 58J50, 81Q15.

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Section: The Reference Hilbert Space and The Resultsmentioning

confidence: 94%

Section: Different Boundary Conditionsmentioning

confidence: 99%

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Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions

Krejčiřı́k¹

2008

ESAIM: COCV

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“…Recently, the first theoretical analysis of the field-free curved waveguide with Dirichlet and Robin requirements on the opposite walls was reported. 84 Further research will reveal bound-state existence and their dependence on the extrapolation length ⌳ and the parameters of the bend for both B = 0 and the nonzero field.…”

Section: Discussionmentioning

confidence: 97%

Analytical and numerical study of a curved planar waveguide with combined Dirichlet and Neumann boundary conditions in a uniform magnetic field

Olendski

Mikhailovska

2008

Phys. Rev. B

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A model of a thin straight strip with a uniformly curved section and with different uniform boundary conditions on the opposite edges subjected to the homogeneous magnetic field B is theoretically analyzed within the framework of the linear Schrödinger equation and is applied to the study of the processes in the bent magnetic multilayers and superconducting films. In particular, for the inner Dirichlet and outer Neumann boundaries, it is shown that bend-induced enhancement of the superconductivity survives in the magnetic field with the order parameter ⌿͑r͒ being pushed stronger and stronger to the Neumann surface with increasing B and, simultaneously, the area where the nucleation of the superconductivity takes place is spread more and more in the straight arms. Various magnetotransport properties of the film such as interference blockade of the supercurrent flow at some special field-dependent temperatures are also discussed with special attention being paid to the formation and evolution of the vortices, which appear as a result of the bend-induced interaction between the different subbands; it is shown, in particular, that the number of vortices decreases with the field and some of them transform into the antivortices. A proof of the very close analogy between two kinds of strips, ͑1͒ pure Dirichlet condition on both edges and ͑2͒ inner Dirichlet and outer Neumann requirements, derived earlier for B = 0 is extended to the case of nonzero fields.

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“…For waveguides with Robin boundary conditions one also has various results in addition to that of Theorem 3.5, one can mention, for instance, [FKr06,Jí07,Kr09,CDN10,OM10,BMT12].…”

Section: Notesmentioning

confidence: 99%

Quantum Waveguides

Exner

Kovařík

2015

Theoretical and Mathematical Physics

143

133

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More information about this series at http://www.springer.com/series/720The series founded in 1975 and formerly (until 2005) entitled Texts and Monographs in Physics (TMP) publishes high-level monographs in theoretical and mathematical physics. The change of title to Theoretical and Mathematical Physics (TMP) signals that the series is a suitable publication platform for both the mathematical and the theoretical physicist. The wider scope of the series is reflected by the composition of the editorial board, comprising both physicists and mathematicians. The books, written in a didactic style and containing a certain amount of elementary background material, bridge the gap between advanced textbooks and research monographs. They can thus serve as basis for advanced studies, not only for lectures and seminars at graduate level, but also for scientists entering a field of research.

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Waveguides with Combined Dirichlet and Robin Boundary Conditions

Cited by 14 publications

References 32 publications

Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions

Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions

Analytical and numerical study of a curved planar waveguide with combined Dirichlet and Neumann boundary conditions in a uniform magnetic field

Quantum Waveguides

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