Semi-classical edge states for the Robin Laplacian

Helffer, B.; Kachmar, A.

doi:10.48550/arxiv.2102.07187

Cited by 2 publications

(2 citation statements)

References 15 publications

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“…Remark 1.3. In [12] the authors analyzed the exponential decay of the eigenfunctions associated with the non positive eigenvalues ω k (h) of the Robin problem. This corresponds to ω = ω k (h).…”

Section: Formentioning

confidence: 99%

Exponential localization of Steklov eigenfunctions on warped product manifolds: the flea on the elephant phenomenon

Daudé¹,

Helffer²,

Nicoleau³

2021

Preprint

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This paper is devoted to the analysis of Steklov eigenvalues and Steklov eigenfunctions on a class of warped product Riemannian manifolds (M, g) whose boundary ∂M consists in two distinct connected components Γ 0 and Γ 1 . First, we show that the Steklov eigenvalues can be divided into two families (λ ± m ) m≥0 which satisfy accurate asymptotics as m → ∞. Second, we consider the associated Steklov eigenfunctions which are the harmonic extensions of the boundary Dirichlet to Neumann eigenfunctions. In the case of symmetric warped product, we prove that the Steklov eigenfunctions are exponentially localized on the whole boundary ∂M as m → ∞. Whenever we add an asymmetric perturbation to a symmetric warped product, we observe a flea on the elephant effect. Roughly speaking, we prove that "half" the Steklov eigenfunctions are exponentially localized on one connected component of the boundary, say Γ 0 , and the other half on the other connected component Γ 1 as m → ∞.

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Section: Formentioning

confidence: 99%

Exponential localization of Steklov eigenfunctions on warped product manifolds: the flea on the elephant phenomenon

Daudé¹,

Helffer²,

Nicoleau³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Estimating the ground state energy of P b γ , leads to information on the critical temperature/critical fields of certain superconductors surrounded by other materials [11,18]. On the other hand, eigenvlaue asymptotics in the singular limit γ → −∞, provide counter examples in the context of spectral geometry [19,4], and has connections to the study of Steklov eigenvalues [20].…”

Section: Introductionmentioning

confidence: 99%

Magnetic perturbations of the Robin Laplacian in the strong coupling limit

Fahs¹

2021

Preprint

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This paper is devoted to the asymptotic analysis of the eigenvalues of the Laplace operator with a uniform magnetic field and Robin boundary condition on a smooth planar domain. We show how the singular limit when the Robin parameter tends to infinity is equivalent to a semi-classical limit involving a small positive parameter h (the semi-classical parameter). The main result is a comparison between the spectrum of the Robin Laplacian with an effective operator defined on the boundary of the domain via the Born-Oppenheimer approximation. More precisely, the n-th eigenvalue of the Robin Laplacian is approximated, modulo O(h 2 ), by the n-th eigenvalue of the effective operator. When the curvature has a unique non-degenerate maximum, the eigenvalue asymptotics displays the contribution of the magnetic field explicitly.

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Semi-classical edge states for the Robin Laplacian

Cited by 2 publications

References 15 publications

Exponential localization of Steklov eigenfunctions on warped product manifolds: the flea on the elephant phenomenon

Exponential localization of Steklov eigenfunctions on warped product manifolds: the flea on the elephant phenomenon

Magnetic perturbations of the Robin Laplacian in the strong coupling limit

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