Low-energy eigenstates in a vanishing magnetic field

TL;DR

This work analyzes the semiclassical spectral problem for the magnetic Laplacian on ${\mathbb R}^2$ in the presence of a magnetic field that vanishes nondegenerately along a smooth curve $\Gamma$. The authors develop a microlocalization framework near $\Gamma$ and a Born–Oppenheimer reduction using operator-valued pseudodifferential calculus and superadiabatic projections, reducing the problem to a finite family of one-dimensional effective Hamiltonians whose spectra approximate the original operator up to $O(h^\infty)$. They establish the existence of discrete low-energy eigenvalues in windows of radius $h^{4/3}$ and provide complete asymptotics, including semiexcited regimes and Bohr–Sommerfeld quantization where applicable; in the bottom of a nondegenerate well, they obtain explicit asymptotics with $h^{4/3}$ scaling and refined corrections. The results extend prior first-eigenvalue analyses to the full low-lying spectrum and give a rigorous framework for magnetic bottles created by vanishing fields, with potential implications for superconductivity models and subRiemannian geometry. Overall, the paper delivers a rigorous, multi-scale spectral description of magnetic Laplacians with vanishing fields, combining microlocal, operator-valued calculus, and detailed asymptotic analysis.

Abstract

This paper is dedicated to the spectral analysis of the semiclassical purely magnetic Laplacian on the plane in the situation where the magnetic field $B$ vanishes nondegenerately on an open smooth curve $Γ$. We prove the existence of a discrete spectrum for energy windows of the scale $h^{4/3}$ and give complete asymptotics in the semiclassical paramater $h$ for eigenvalues in such windows. Our strategy relies on the microlocalization of the corresponding eigenfunctions close to the zero locus $Γ$ and on the implementation of a Born-Oppenheimer strategy through the use of operator-valued pseudodifferential calculus and superadiatic projectors. This allows us to reduce our spectral analysis to that of effective semiclassical pseudodifferential operators in dimension 1 and apply the well-known semiclassical techniques à la Helffer-Sjöstrand.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2: DR,Mon
• Theorem 1.3: HL, Theorem 1.5
• Definition 1.4: FLTRVN
• Theorem 1.5
• Corollary 1.7
• Corollary 1.9
• Remark 2.6
• Theorem 2.7: Persson, Pe
• proof : Proof of Theorem \ref{['thm:spectrediscret']}