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Semi-classical eigenvalue estimates under magnetic steps (Former title: Hearing the shape of a magnetic edge in the semiclassical limit)

Wafaa Assaad, Bernard Helffer, Ayman Kachmar

Abstract

We establish accurate eigenvalue asymptotics and, as a by-product, sharp estimates of the splitting between two consecutive eigenvalues, for the Dirichlet magnetic Laplacian with a non-uniform magnetic field having a jump discontinuity along a smooth curve. The asymptotics hold in the semiclassical limit which also corresponds to a large magnetic field limit, and is valid under a geometric assumption on the curvature of the discontinuity curve.

Semi-classical eigenvalue estimates under magnetic steps (Former title: Hearing the shape of a magnetic edge in the semiclassical limit)

Abstract

We establish accurate eigenvalue asymptotics and, as a by-product, sharp estimates of the splitting between two consecutive eigenvalues, for the Dirichlet magnetic Laplacian with a non-uniform magnetic field having a jump discontinuity along a smooth curve. The asymptotics hold in the semiclassical limit which also corresponds to a large magnetic field limit, and is valid under a geometric assumption on the curvature of the discontinuity curve.

Paper Structure

This paper contains 28 sections, 18 theorems, 399 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Case $a=-1$
  3. Case $-1<a<0$
  4. Case $a=0$
  5. Case $0<a<1$
  6. Comparison with earlier situations
  7. Fiber operators
  8. Band functions
  9. Properties of band functions/states
  10. Moments
  11. 1D model involving the curvature
  12. Upper bound
  13. Functions localized near the magnetic edge
  14. Localization hypotheses
  15. Rescaled functions and tangential estimates
  16. ...and 13 more sections

Key Result

Theorem 1.2

Let $n\in\mathbb{N}^*$ and $\mathbf a=(1,a)$ with $-1<a<0$ . Under Assumption kmax, the $n$'th eigenvalue $\lambda_n(h)$ of $\mathcal{P}_{h}$, defined in eq:P, satisfies as $h\rightarrow 0$, where $\beta_a$, $c_2(a)$ and $M_3(a)$ are the spectral quantities introduced in eq:beta0 and eq:main-ct.

Figures (1)

  • Figure 1: The curve $\Gamma$ transversally cuts $\partial\Omega$ at two points and splits $\Omega$ into two regions, $\Omega_1$ and $\Omega_2$.

Theorems & Definitions (41)

  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 31 more