Semiclassical tunneling and magnetic flux effects on the circle

Bonnaillie‐Noël, Virginie; Hérau, Frédéric; Raymond, Nicolas

doi:10.4171/jst/177

Cited by 9 publications

(23 citation statements)

References 11 publications

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“…We follow the exposition of [14, §4.5] (see also [35]) but note that the formulas are established only for an example. In this paper, we will also use in many places the presentation of [2]. Because there are two geodesics between the two wells the discussion will depend on the comparison between the lengths of these two geodesics.…”

Section: 22mentioning

confidence: 99%

Tunneling for the Robin Laplacian in smooth planar domains

Helffer

Kachmar

Raymond

2016

Commun. Contemp. Math.

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article number: 1650030International audienceWe study the low-lying eigenvalues of the semiclassical Robin Laplacian in a smooth planar domain symmetric with respect to an axis. In the case when the curvature of the boundary of the domain attains its maximum at exactly two points away from the axis of symmetry, we establish an explicit asymptotic formula for the splitting of the first two eigenvalues. This is a rigorous derivation of the semiclassical tunneling effect induced by the domain's geometry. Our approach is close to the Born-Oppenheimer one and yields, as a byproduct, a Weyl formula of independent interest

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Section: 22mentioning

confidence: 99%

Tunneling for the Robin Laplacian in smooth planar domains

Helffer

Kachmar

Raymond

2016

Commun. Contemp. Math.

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“…This proves that the presence of the magnetic field amplifies the tunneling effect and that it has even another order of magnitude with respect to ε when f 0 L ∈ {π/2 + kπ , k ∈ Z}. The strategies in the previous works [4], [9] were developed (in particular) to establish the improvement of a geometric tunneling by the presence of the magnetic field. As far as we know, Theorem 1.3 is the first result proving such an effect for a magnetic two dimensional model.…”

Section: Remark 14mentioning

confidence: 80%

“…For ellipses, only a conjecture has been provided in [2] and numerically checked. This conjecture was suggested by a formal dimensional reduction and the analysis in [4], [14].…”

Section: Motivation and Contextmentioning

confidence: 91%

“…We tackle an elementary geometric situation which can be analyzed via a reduction to dimension one (in the shrinking limit ε → 0, which turns out to be of semiclassical nature), in the case when the magnetic field is a pure flux. The behavior of the spectral gap λ 2 (ε) − λ 1 (ε) in the limit ε → 0 can be established via the methods developed in recent papers (see [4], [9]). Our aim is to explain how these previous contributions can be combined to establish a tunneling result involving a pure magnetic field in a limit of semiclassical nature: such results are indeed rather rare in the literature.…”

Section: Motivation and Contextmentioning

confidence: 99%

“…where [p, q] denotes the arc joining p and q in C counter-clockwise. The indices u and d refer to the up and down parts of C. Using a rescaling and Theorem 1.4 in [4] (see also [14]), we have…”

Section: The Effective Operatormentioning

confidence: 99%

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Tunnel effect in a shrinking shell enlacing a magnetic field

Kachmar

Raymond

2019

Rev. Mat. Iberoam.

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Let C be a smooth planar curve. We assume that C is simple, closed, smooth, symmetric with respect to an axis and its curvature attains its minimum at exactly two points away from the axis of symmetry. In a tubular neighborhood about C, we study the Laplace operator with a magnetic flux and mixed boundary conditions. As the thickness of the domain tends to 0, we establish an explicit asymptotic formula for the splitting of the first two eigenvalues (tunneling effect).

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Flux and symmetry effects on quantum tunneling

Helffer,

Kachmar,

Sundqvist

2024

Math. Ann.

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Motivated by the analysis of the tunneling effect for the magnetic Laplacian, we introduce an abstract framework for the spectral reduction of a self-adjoint operator to a hermitian matrix. We illustrate this framework by three applications, firstly the electro-magnetic Laplacian with constant magnetic field and three equidistant potential wells, secondly a pure constant magnetic field and Neumann boundary condition in a smoothed triangle, and thirdly a magnetic step where the discontinuity line is a smoothed triangle. Flux effects are visible in the three aforementioned settings through the occurrence of eigenvalue crossings. Moreover, in the electro-magnetic Laplacian setting with double well radial potential, we rule out an artificial condition on the distance of the wells and extend the range of validity for the tunneling approximation recently established in Fefferman et al. (SIAM J Math Anal 54: 1105–1130, 2022), Helffer & Kachmar (Pure Appl Anal, 2024), thereby settling the problem of electro-magnetic tunneling under constant magnetic field and a sum of translated radial electric potentials.

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Semiclassical tunneling and magnetic flux effects on the circle

Cited by 9 publications

References 11 publications

Tunneling for the Robin Laplacian in smooth planar domains

Tunneling for the Robin Laplacian in smooth planar domains

Tunnel effect in a shrinking shell enlacing a magnetic field

Flux and symmetry effects on quantum tunneling

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