Topologically nontrivial quantum layers

Carron, Gilles; Exner, Pavel; Krejčiřı́k, David

doi:10.1063/1.1635998

Cited by 69 publications

(99 citation statements)

References 9 publications

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“…we proceed first with a change of variables in Ω δ with small δ. The computations below are very similar to those performed in [3] for a different problem.…”

Section: Auxiliary Estimatessupporting

confidence: 67%

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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“…we proceed first with a change of variables in Ω δ with small δ. The computations below are very similar to those performed in [3] for a different problem.…”

Section: Auxiliary Estimatessupporting

confidence: 67%

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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“…The regime with small angle limit has been studied in [5] and more recently in [10,11]. The question of waveguides with corner arises naturally because it is studied for smooth waveguides in [13,6,7] where we learn, among other things, that curvature induces bound states below the essential spectrum. The idea is that a corner can be seen as an infinite curvature.…”

Section: Motivations and Related Questionsmentioning

confidence: 99%

“…One can pass from Circ(θ) to Circ(θ) by the change of variables r = ρ cos α − 1, z = ρ sin α, which links those two domains without the cartesian domain. convergence rate for 1 [0] convergence rate for 2 [0] convergence rate for 4 [0] convergence rate for 5 [0] convergence rate for 6 [0] n (tan θ) to the two first terms of the theoretical asymptotics on the aperture θ (in degrees). The black line represents the value 4/3.…”

Section: Remarkmentioning

confidence: 99%

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Dirichlet eigenvalues of cones in the small aperture limit

Ourmières-Bonafos¹

2014

J. Spectr. Theory

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We are interested in finite cones of fixed height 1 parametrized by their opening angle. We study the eigenpairs of the Dirichlet Laplacian in such domains when their apertures tend to 0. We provide multi-scale asymptotics for eigenpairs associated with the lowest eigenvalues of each fiber of the Dirichlet Laplacian. In order to do this, we investigate the family of their one-dimensional Born-Oppenheimer approximations. The eigenvalue asymptotics involves powers of the cube root of the aperture, while the eigenfunctions include simultaneously two scales.

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“…The evidently more complicated case of layers, i.e. Ω is a tubular neighbourhood about a complete non-compact surface in R 3 , was investigated in [19,20,38,12,64].…”

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confidence: 99%

On the Spectrum of Curved Planar Waveguides

Krejčiřı́k¹,

Křı́ž²

2005

Publ. Res. Inst. Math. Sci.

102

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The spectrum of the Laplace operator in a curved strip of constant width built along an infinite plane curve, subject to three different types of boundary conditions (Dirichlet, Neumann and a combination of these ones, respectively), is investigated. We prove that the essential spectrum as a set is stable under any curvature of the reference curve which vanishes at infinity and find various sufficient conditions which guarantee the existence of geometrically induced discrete spectrum. Furthermore, we derive a lower bound to the distance between the essential spectrum and the spectral threshold for locally curved strips. The paper is also intended as an overview of some new and old results on spectral properties of curved quantum waveguides.

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Topologically nontrivial quantum layers

Cited by 69 publications

References 9 publications

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

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

Dirichlet eigenvalues of cones in the small aperture limit

On the Spectrum of Curved Planar Waveguides

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