Geometry and Spectrum in 2D Magnetic Wells

Raymond, Nicolas; Ngọc, San Vũ

doi:10.5802/aif.2927

Cited by 50 publications

(80 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 2.4. The case of purely magnetic Schrödinger operators when the magnetic field has a global non-degenerate minimum has been treated in [15,36].…”

Section: Main Steps In the Proofmentioning

confidence: 99%

Low lying spectral gaps induced by slowly varying magnetic fields

Cornean

Helffer

Purice

2017

Journal of Functional Analysis

View full text Add to dashboard Cite

We consider a periodic Schr\"odinger operator in two dimensions perturbed by a weak magnetic field whose intensity slowly varies around a positive mean. We show in great generality that the bottom of the spectrum of the corresponding magnetic Schr\"odinger operator develops spectral islands separated by gaps, reminding of a Landau-level structure. First, we construct an effective Hofstadter-like magnetic matrix which accurately describes the low lying spectrum of the full operator. The construction of this effective magnetic matrix does not require a gap in the spectrum of the non-magnetic operator, only that the first and the second Bloch eigenvalues do not cross but their ranges might overlap. The crossing case is more difficult and will be considered elsewhere. Second, we perform a detailed spectral analysis of the effective matrix using a gauge-covariant magnetic pseudo-differential calculus adapted to slowly varying magnetic fields. As an application, we prove in the overlapping case the appearance of spectral islands separated by gaps.Comment: 55 pages, to appear in J. Funct. Ana

show abstract

“…Remark 2.4. The case of purely magnetic Schrödinger operators when the magnetic field has a global non-degenerate minimum has been treated in [15,36].…”

Section: Main Steps In the Proofmentioning

confidence: 99%

Low lying spectral gaps induced by slowly varying magnetic fields

Cornean

Helffer

Purice

2017

Journal of Functional Analysis

View full text Add to dashboard Cite

show abstract

“…Under the assumption (2.12), the known spectral asymptotics (which are actually the same in this case) of the Dirichlet and Neumann eigenvalues will lead us to the asymptotics given in the righthand side of (2.13). Under the additional assumption that inf x∈Ω |B 0 (x)| is attained at a unique minimum in Ω and that this minimum is non degenerate, a complete asymptotics of λ N (tA 0 ) can be given (see Helffer-Mohamed [15], Helffer-Kordyukov [13,14], Raymond-Vu Ngoc [23] ) and the monotonicity/strong diamagnetism property holds for large values of t (see Chapter 3 in [4]). Hence the definition of H c 3 (κ) is clear in this case.…”

Section: 2mentioning

confidence: 99%

The Ginzburg–Landau Functional with Vanishing Magnetic Field

Helffer

Kachmar

2015

Arch Rational Mech Anal

View full text Add to dashboard Cite

Abstract. We study the infimum of the Ginzburg-Landau functional in the case of a vanishing external magnetic field in a two dimensional simply connected domain. We obtain an energy asymptotics which is valid when the Ginzburg-Landau parameter is large and the strength of the external field is comparable with the third critical field. Compared with the known results when the external magnetic field does not vanish, we show in this regime a concentration of the energy near the zero set of the external magnetic field. Our results complete former results obtained by K. Attar and X-B. Pan-K-H. Kwek.

show abstract

“…This fact was noticed, for instance, in the papers by Helffer and Morame [18,19] where numerous techniques have been developed to analyze the magnetic Laplacian and its eigenfunctions. Even more recently in [16,34,17], in cases without boundary, subtle localization properties of the magnetic eigenfunctions have played a fundamental role in the semiclassical spectral theory (and we will meet again this aspect in the nonlinear context). In cases with boundaries, the Robin condition is physically motivated by inhomogeneous superconductors (see for instance the linear and nonlinear contributions by Kachmar [21,22,23,20]): in this context, the Robin condition is sometimes called "de Gennes condition".…”

Section: Some Perspectives 39mentioning

confidence: 81%

Semiclassical Sobolev constants for the electro-magnetic Robin Laplacian

Fournais¹,

Treust²,

Raymond³

et al. 2017

J. Math. Soc. Japan

Self Cite

View full text Add to dashboard Cite

Abstract. This paper is devoted to the asymptotic analysis of the optimal Sobolev constants in the semiclassical limit and in any dimension. We combine semiclassical arguments and concentration-compactness estimates to tackle the case when an electro-magnetic field is added as well as a smooth boundary carrying a Robin condition. As a byproduct of the semiclassical strategy, we also get exponentially weighted localization estimates of the minimizers.

show abstract

Geometry and Spectrum in 2D Magnetic Wells

Cited by 50 publications

References 28 publications

Low lying spectral gaps induced by slowly varying magnetic fields

Low lying spectral gaps induced by slowly varying magnetic fields

The Ginzburg–Landau Functional with Vanishing Magnetic Field

Semiclassical Sobolev constants for the electro-magnetic Robin Laplacian

Contact Info

Product

Resources

About