On the magnetic Dirichlet to Neumann operator on the disk -- strong diamagnetism and strong magnetic field limit--

TL;DR

The article resolves the Chakradhar–Gittins–Habib–Peyerimhoff conjecture by establishing a precise large-$b$ expansion for the ground state energy of the magnetic Dirichlet-to-Neumann map on the unit disk, showing $\lambda^{DN}(b)=\alpha\sqrt{b}-\frac{\alpha^2+2}{6}+O(b^{-1/2})$ with $\alpha$ the negative root of $D_{1/2}$. It develops a detailed spectral analysis via magnetic Steklov eigenvalues $\lambda_n(b)$, characterizes their intersections $z_n$, proves strong diamagnetism (monotonicity of $\lambda^{DN}(b)$ in $b$), and derives the asymptotics of $z_n$ and $\lambda^{DN}$. A half-plane model and variational characterization are used to connect the disk problem to Neumann-type De Gennes theory and to extend insights toward curvature effects in general domains. The results provide a rigorous bridge between magnetic boundary phenomena, special function theory (Kummer, Laguerre, parabolic cylinder functions), and asymptotic spectral analysis with implications for diamagnetism and boundary curvature corrections.

Abstract

Inspired by a paper by T. Chakradhar, K. Gittins, G. Habib and N. Peyerimhoff, we analyze their conjecture that the ground state energy of the magnetic Dirichlet-to-Neumann operator on the disk tends to $+\infty$ as the magnetic field tends to $+\infty$. This is an important step towards the analysis of the curvature effect in the case of general domains in $\mathbb R^2$.

Figures (5)

• Figure 1: The magnetic Steklov eigenvalues $\lambda_n (b)$ (left) and the ground state energy $\lambda^{DN}(b)$ (right).
• Figure 2: Graph of $f(\xi)$.
• Figure 3: Graph proposed by Saint-James.
• Figure 4: Graph of the function $D_\frac{1}{2} (x)$.
• Figure 5: Graph of the function $f_1(\xi)$.

Theorems & Definitions (41)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Remark 2.2
• Proposition 3.1
• proof
• Proposition 4.1
• proof