Discrete spectrum of interactions concentrated near conical surfaces

Ourmières-Bonafos, Thomas; Pankrashkin, Konstantin

doi:10.1080/00036811.2017.1325472

Cited by 15 publications

(13 citation statements)

References 24 publications

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“…Recently, in [29], it has been proved that the infiniteness of bound states and the logarithmic accumulation to the threshold of the essential spectrum still hold for conical layers constructed around any smooth reference conical surface (by smooth reference conical surface, we mean that the surface is smooth except in its vertex).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Dirichlet Spectrum of the Fichera Layer

Dauge

Lafranche

Ourmières-Bonafos

2018

Integr. Equ. Oper. Theory

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We investigate the spectrum of the three-dimensional Dirichlet Laplacian in a prototypal infinite polyhedral layer, that is formed by three perpendicular quarter-plane walls of constant width joining each other. Alternatively, this domain can be viewed as an octant from which another "parallel" octant is removed. It contains six edges (three convex and three non-convex) and two corners (one convex and one non-convex). It is a canonical example of non-smooth conical layer. We name it after Fichera because near its non-convex corner, it coincides with the famous Fichera cube that illustrates the interaction between edge and corner singularities. This domain could also be called an octant layer.We show that the essential spectrum of the Laplacian on such a domain is a half-line and we characterize its minimum as the first eigenvalue of the two-dimensional Laplacian on a broken guide. By a Born-Oppenheimer type strategy, we also prove that its discrete spectrum is finite and that a lower bound is given by the ground state of a special Sturm-Liouville operator. By finite element computations taking singularities into account, we exhibit exactly one eigenvalue under the essential spectrum threshold leaving a relative gap of 3%. We extend these results to a variant of the Fichera layer with rounded edges (for which we find a very small relative gap of 0.5%), and to a three-dimensional cross where the three walls are full thickened planes.2010 Mathematics Subject Classification. 35J05, 35P15, 35Q40, 81Q10, 65M60.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…i) The bottom of the essential spectrum is driven by the first eigenvalue of associated two-dimensional quantum wave guides; ii) The number of independent bound states is finite. This contrasts with the family of smooth conical layers, denoted here as C, investigated in [29], in which the Dirichlet Laplacian satisfies:…”

Section: Resultsmentioning

confidence: 99%

Dirichlet Spectrum of the Fichera Layer

Dauge

Lafranche

Ourmières-Bonafos

2018

Integr. Equ. Oper. Theory

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“…The results were then extended to δ-potentials supported by non-circular conical surfaces in [9,18], and we refer to [5,6,11,15,19] for the discussion of other types of differential operators in conical geometries. The goal of the present paper is to Key words and phrases.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Spectral asymptotics for δ-interactions on sharp cones

Ourmières-Bonafos

Pankrashkin

Pizzichillo

2018

Journal of Mathematical Analysis and Applications

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We investigate the spectrum of three-dimensional Schrödinger operators with δ-interactions of constant strength supported on circular cones. As shown in earlier works, such operators have infinitely many eigenvalues below the threshold of the essential spectrum. We focus on spectral properties for sharp cones, that is when the cone aperture goes to zero, and we describe the asymptotic behavior of the eigenvalues and of the eigenvalue counting function. A part of the results are given in terms of numerical constants appearing as solutions of transcendental equations involving modified Bessel functions.

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“…In this perspective, more precise analysis of various particular cases deserves attention. Examples of layers treated so far include mildly curved layers [BEGK01,EKr01], conical layers [DOR15,ET10,OP17], and so-called octant (or Fichera) layers [DLO17].…”

Section: State Of the Art And Motivationmentioning

confidence: 99%

Spectral Asymptotics of the Dirichlet Laplacian on a Generalized Parabolic Layer

Exner

Lotoreichik

2020

Integr. Equ. Oper. Theory

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We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian H on an unbounded, radially symmetric (generalized) parabolic layer P ⊂ R 3 . It was known before that H has an infinite number of eigenvalues below the threshold of its essential spectrum. In the present paper, we find the discrete spectrum asymptotics for H by means of a consecutive reduction to the analogous asymptotic problem for an effective one-dimensional Schrödinger operator on the half-line with the potential the behaviour of which far away from the origin is determined by the geometry of the layer P at infinity.tors.

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Discrete spectrum of interactions concentrated near conical surfaces

Cited by 15 publications

References 24 publications

Dirichlet Spectrum of the Fichera Layer

Dirichlet Spectrum of the Fichera Layer

Spectral asymptotics for δ-interactions on sharp cones

Spectral Asymptotics of the Dirichlet Laplacian on a Generalized Parabolic Layer

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