On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator in non-simply connected domains

On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator

We discuss the results of a recent paper by Ekholm, Kovařík and Portmann in connection with a question of C. Guillarmou about the semiclassical expansion of the lowest eigenvalue of the Pauli operator with Dirichlet conditions. We exhibit connections with the properties of the torsion function in mechanics, the exit time of a Brownian motion and the analysis of the low eigenvalues of some Witten Laplacian.

On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator III: Magnetic fields that change sign

We consider the semi-classical Dirichlet Pauli operator in bounded connected domains in the plane. Rather optimal results have been obtained in previous papers by Ekholm-Kovařík-Portmann and Helffer-Sundqvist for the asymptotics of the ground state energy in the semi-classical limit when the magnetic field has constant sign. In this paper, we focus on the case when the magnetic field changes sign. We show, in particular, that the ground state energy of this Pauli operator will be exponentially small as the semi-classical parameter tends to zero and give lower bounds and upper bounds for this d…

On the semiclassical spectrum of the Dirichlet-Pauli operator

This paper is devoted to semiclassical estimates of the eigenvalues of the Pauli operator on a bounded open set whose boundary carries Dirichlet conditions. Assuming that the magnetic field is positive and a few generic conditions, we establish the simplicity of the eigenvalues and provide accurate asymptotic estimates involving Segal-Bargmann and Hardy spaces associated with the magnetic field.

Semiclassical spectrum of the Dirichlet-Pauli operator on an annulus

This paper is devoted to the semiclassical analysis of the spectrum of the Dirichlet-Pauli operator on an annulus. We assume that the magnetic field is strictly positive and radial. We give an explicit asymptotic expansion at the first order of the lowest eigenvalues of this operator in the semiclassical limit. In particular, we exhibit the Aharonov-Bohm effect that has been revealed, for constant magnetic field, in a recent paper by Helffer and Sundqvist.

区域上Pauli算子的近似零模

Approximate Zero Modes for the Pauli Operator on a Region

设$\mathcal{P}_{Ω,tA}$表示有界开区域$Ω\subset\mathbb{R}^2$上的Pauli算子，其Dirichlet边界条件和磁势$A$由某个$t>0$标度.假设相应的磁场$B=\mathrm{curl}\,A$满足$B\in L\log L(Ω)\cap C^α(Ω_0)$，其中$α>0$和$Ω_0$是全测度$Ω$的开子集（注意，Orlicz空间$L\log L(Ω)$包含任何$p>1$的$L^p(Ω)$）。令$\mathsf{N}_{Ω,tA}(λ)$表示相应的特征值计数函数。我们建立了强场渐近公式\[ \mathsf{N}_{Ω,tA}(λ(t))=\frac{t}{2π}\int_Ω\lvert B(x)\rvert\,dx\;+o(t) \]为$t\to+\infty$，当$λ(t)=Ce^{-ct^σ}$为某些$σ\in(0,1)$和$c,C>0$时.相应的本征函数可以看作是$\mathbb{R}^2$上Pauli算子的Aharonov-Casher零模的局部化形式。

Spectral properties of 2D Pauli operators with almost periodic electromagnetic fields

We consider a 2D Pauli operator with almost periodic field $b$ and electric potential $V$. First, we study the ergodic properties of $H$ and show, in particular, that its discrete spectrum is empty if there exists an almost periodic magnetic potential which generates the magnetic field $b - b_{0}$, $b_{0}$ being the mean value of $b$. Next, we assume that $V = 0$, and investigate the zero modes of $H$. As expected, if $b_{0} \neq 0$, then generically $\operatorname{dim} \operatorname{Ker} H = \infty$. If $b_{0} = 0$, then for each $m \in {\mathbb N} \cup \{ \infty \}$, we construct almost peri…