Robin eigenvalues on domains with peaks

Kovařík, Hynek; Pankrashkin, Konstantin

doi:10.1016/j.jde.2019.02.016

Cited by 7 publications

(5 citation statements)

References 15 publications

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“…The analysis is in the spirit of the Born-Oppenheimer approximation, see e.g. [13,Part 3], with M ε ′ ,a being an "effective operator", and it is essentially an adaptation of the constructions of the earlier paper [9] used for the study of Robin eigenvalues in domains with peaks. We then show in Proposition 9 that the eigenvalues of Q ε are close to those of T ε , which finishes the proof of Theorem 1.…”

Section: Scheme Of the Proofmentioning

confidence: 99%

“…Our proof is variational and based on the min-max principle, and its main ingredient is a kind of asymptotic separation of the variables x 1 and x ′ , which is quite similar to [8]. On the other hand, the analysis in the x ′ -variable becomes much more involved and uses some results on the Robin eigenvalues of balls from the earlier paper [9]. Various proof steps are explained in greater detail in Subsection 2.3 below.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotics of Robin eigenvalues on sharp infinite cones

Pankrashkin¹,

Vogel²

2022

Preprint

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For ε > 0 and n ∈ N consider the infinite coneand the operator Q α ε acting as the Laplacian u → −∆u on Ω ε with the Robin boundary condition ∂ ν u = αu at ∂Ω ε , where ∂ ν is the outward normal derivative and α > 0. It is known from numerous earlier works that the essential spectrum of Q α ε is [−α 2 , +∞) and that the discrete spectrum is finite for n = 1 and infinite for n ≥ 2, but the behavior of individual eigenvalues with respect to the geometric parameter ε was only addressed for n = 1 so far. In the present work we consider arbitrary n ≥ 2 and look at the spectral asymptotics as ε becomes small, i.e. as the cone becomes "sharp" and collapses to its central axis. Our main result is as follows: if n ≥ 2, α > 0 and j ∈ N are fixed, then the jth eigenvalue E j (Q α ε ) of Q α ε behaves as

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Section: Scheme Of the Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Asymptotics of Robin eigenvalues on sharp infinite cones

Pankrashkin¹,

Vogel²

2022

Preprint

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“…For a domain with a cusp, the resulting operator is not necessarily self-adjoint. If the cusp is, roughly, less sharp than quadratic, then the form is bounded from below, and the spectrum is discrete (see [16,22,24] and, e.g., [11] for a recent study of the corresponding eigenvalue sequence itself). But, as it was shown in [22,24], the nature of the problem operator may become completely different, as it may lose its semi-boundedness, if the cusp is sharper than quadratic, see also [4, § 5].…”

Section: State Of Artmentioning

confidence: 99%

Plummeting and blinking eigenvalues of the Robin Laplacian in a cuspidal domain

Назаров¹,

Popoff²,

Taskinen³

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We consider the Robin Laplacian in the domains Ω and Ω ε , ε > 0, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient a is large enough, the spectrum of the problem in Ω is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain Ω ε is discrete. However, our results reveal the strange behavior of the discrete spectrum as the blunting parameter ε tends to 0: we construct asymptotic forms of the eigenvalues and detect families of "hardly movable" and "plummeting" ones. The first type of the eigenvalues do not leave a small neighborhood of a point for any small ε > 0 while the second ones move at a high rate O(| ln ε|) downwards along the real axis R to −∞. At the same time, any point λ ∈ R is a "blinking eigenvalue", i.e., it belongs to the spectrum of the problem in Ω ε almost periodically in the | ln ε|-scale. Besides standard spectral theory, we use the techniques of dimension reduction and self-adjoint extensions to obtain these results.

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“…This is in contrast with the one-sided Robin problems for the Laplacian in a domain surrounded by Γ, for which the cusp is not allowed to be very sharp: see e.g. [17] for the study of the eigenvalues and [19,20] for the issues concerning the definition of the operator.…”

Section: Introductionmentioning

confidence: 99%

Strong coupling asymptotics for $δ$-interactions supported by curves with cusps

Flamencourt,

Pankrashkin

2019

Preprint

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Let Γ ⊂ R 2 be a simple closed curve which is smooth except at the origin, at which it has a power cusp and coincides with the curve |x 2 | = x p 1 for some p > 1. We study the eigenvalues of the Schrödinger operator H α with the attractive δ-potential of strength α > 0 supported by Γ, which is defined by its quadratic formwhere ds stands for the one-dimensional Hausdorff measure on Γ.It is shown that if n ∈ N is fixed and α is large, then the welldefined nth eigenvalue E n (H α ) of H α behaves as, where the constants E n > 0 are the eigenvalues of an explicitly given one-dimensional Schrödinger operator determined by the cusp, and η > 0. Both main and secondary terms in this asymptotic expansion are different from what was observed previously for the cases when Γ is smooth or piecewise smooth with non-zero angles.with δ being the Dirac distribution and α > 0 being the coupling constant. Such operators describe the motion of particles confined to the graph Γ but allowing for a quantum tunneling between its different

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Robin eigenvalues on domains with peaks

Cited by 7 publications

References 15 publications

Asymptotics of Robin eigenvalues on sharp infinite cones

Asymptotics of Robin eigenvalues on sharp infinite cones

Plummeting and blinking eigenvalues of the Robin Laplacian in a cuspidal domain

Strong coupling asymptotics for $δ$-interactions supported by curves with cusps

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