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

Krejčiřı́k, David

doi:10.1051/cocv:2008035

Cited by 35 publications

(51 citation statements)

References 27 publications

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“…We remark that a presence of the first mean curvature in eigenvalue asymptotics has been recently observed in related problems, see [24], [25] and [34].…”

Section: Introductionsupporting

confidence: 54%

Asymptotic spectral analysis in colliding leaky quantum layers

Kondej

Krejčiřı́k

2017

Journal of Mathematical Analysis and Applications

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We consider the Schrödinger operator with a complex delta interaction supported by two parallel hypersurfaces in the Euclidean space of any dimension. We analyse spectral properties of the system in the limit when the distance between the hypersurfaces tends to zero. We establish the norm-resolvent convergence to a limiting operator and derive first-order corrections for the corresponding eigenvalues.

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“…We remark that a presence of the first mean curvature in eigenvalue asymptotics has been recently observed in related problems, see [24], [25] and [34].…”

Section: Introductionsupporting

confidence: 54%

Asymptotic spectral analysis in colliding leaky quantum layers

Kondej

Krejčiřı́k

2017

Journal of Mathematical Analysis and Applications

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“…[12,13,20,21]. The reduced operator −∆ S − αK appeared already in [21] in the study of suitable Laplacians in thin neighbordnood of hypersurfaces, and the results from Section 8 provide improvements of the asymptotics given in [21, Theorem 1.1], under the respective geometric assumptions.…”

Section: Introductionmentioning

confidence: 89%

An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter

Pankrashkin

Popoff

2016

Journal de Mathématiques Pures et Appliquées

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Abstract. We consider the Laplacian on a class of smooth domains Ω ⊂ R ν , ν ≥ 2, with attractive Robin boundary conditions:where n is the outer unit normal, and study the asymptotics of its eigenvalues Ej(Q Ω α ) as well as some other spectral properties for α → +∞ We work with both compact domains and non-compact ones with a suitable behavior at infinity. For domains with compact C 2 boundaries and fixed j, we show thatwhere µj(α) is the j th eigenvalue, as soon as it exists, of −∆S−(ν−1)αH with (−∆S) and H being respectively the positive Laplace-Beltrami operator and the mean curvature on ∂Ω. Analogous results are obtained for a class of domains with non-compact boundaries.In particular, we discuss the existence of eigenvalues in non-compact domains and the existence of spectral gaps for periodic domains. We also show that the remainder estimate can be improved under stronger regularity assumptions. The effective Hamiltonian −∆S − (ν − 1)αH enters the framework of semi-classical Schrödinger operators on manifolds, and we provide the asymptotics of its eigenvalues in the limit α → +∞ under various geometrical assumptions. In particular, we describe several cases for which our asymptotics provides gaps between the eigenvalues of Q IntroductionLet Ω ⊂ R ν , ν ≥ 2, be an open set with a sufficiently regular boundary S := ∂Ω. For α ∈ R, denote by Q Ω α the operator Q Ω α u = −∆u on the functions u defined in Ω and satisfying the Robin boundary condition ∂u ∂nwhere n is the outer unit normal at S. More precisely,where dS stands for the (ν − 1)-dimensional Hausdorff measure on S, which is closed and semibounded from below under suitable assumptions (e.g. if S is compact or with a suitable behavior at infinity, see below), and we denote by E Ω j (Q Ω α ) the j th eigenvalue of Q Ω α below the bottom of the essential spectrum, as soon as it exists. The aim of the paper is to obtain new results on the asymptotics of the eigenvalues as α tends to +∞.The problem appears in various applications, such as reaction-diffusion processes [25] and the enhanced surface superconductivity [14], and the related questions were already 1 2 discussed in the previous works by various authors. Let us present briefly the state of art for compact domains. It was shown in [25,26] that for piecewise smooth Liptschotz domain one has E 1 (Q Ω α ) = −C Ω α 2 + o(α 2 ) as α → +∞, where C Ω ≥ 1 is a constant depending on the geometric properties of Ω. In particular, C Ω = 1 for C 1 domains, see [5,27]. More detailed asymptotic expansions for some specific non-smooth domains were considered in [17,26,31]. As for smooth domains, a more detailed result was obtained first in [9,30] for ν = 2 and then in [32] for any ν ≤ 2: if the domain is C 3 and j ∈ N is fixed, thenwhere H max is the maximum of the mean curvature H of the boundary (the exact definition will be recalled below). Remark that this asymptotics together with isompermetric inequalities for the mean curvature have played an important role for the so-called reverse Faber-Krahn inequality, ...

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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

142

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

Cited by 35 publications

References 27 publications

Asymptotic spectral analysis in colliding leaky quantum layers

Asymptotic spectral analysis in colliding leaky quantum layers

An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter

Quantum Waveguides

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