Sharp asymptotic estimates for eigenvalues of Aharonov-Bohm operators with varying poles (Q5963593)

Let \(\Omega\subset {\mathbb R}^2\) be a bounded, open and simply connected domain such that \(0\in\Omega\). The authors investigate the behavior of eigenvalues for a magnetic Aharonov-Bohm operator \((i\nabla + A_a)^2\) on \(L^2(\Omega)\) with Dirichlet boundary conditions and with half-integer circulation: \[ A_a(x)=\frac{1}{2}\left(\frac{-(x_2-a_2)}{(x_1-a_1)^2+(x_2-a_2)^2},\frac{x_1-a_1}{(x_1-a_1)^2+(x_2-a_2)^2}\right) \] for \(a=(a_1,a_2)\in{\mathbb R}^2\), \(x=(x_1,x_2) \in{\mathbb R}^2\backslash\{a\}\). Nodal domains of eigenfunctions of such operators are strongly related to spectral minimal partitions of the Dirichlet Laplacian. Let \(n_0 \geq 1\) be such that the \(n_0\)-th eigenvalue \(\lambda^0 = \lambda^0_{n_0}\) of \((i\nabla + A_0)^2\) is simple with associated eigenfunctions having in \(0\) a zero of order \(k/2\) with \(k \in {\mathbb N}\) odd. For \(a \in\Omega\), let \(\lambda^a = \lambda^a_{n_0}\) be the \(n_0\)-th eigenvalue of \((i\nabla + A_a)^2\). Let \(\tau\) be the half-line tangent to a nodal line of eigenfunctions associated to \(\lambda_0\) ending at \(0\). It is proved that \( (\lambda_0-\lambda_a)/|a|^k \to \operatorname{const}\) as \(a\to 0\) with \(a\in\tau\). The sharp coefficient of this asymptotics is detected; it can be characterized in terms of the limit profile of a blow-up sequence obtained by a suitable scaling of approximating eigenfunctions. As a consequence, some conjectures arising from numerical evidence in the preexisting literature are verified theoretically. The proof relies on an Almgren-type monotonicity argument for magnetic operators together with a sharp blow-up analysis.