Tunnel effect in a shrinking shell enlacing a magnetic field (Q2280521)

Let \(C\) be a simple, smooth and closed curve in \({\mathbb R}^2\). It is also assumed that \(C\) is symmetric with respect to an axis and its curvature attains its minimum at exactly two points away from the axis of symmetry. The tubular domain with thickness \(\varepsilon 0\) is the set \(\Omega_\varepsilon=\{x\in\Omega:0 \mathrm{ dist}(x,C)\varepsilon\}\). The authors consider the Laplace operator with magnetic field \( \mathcal{L}_\varepsilon = -(\nabla-i \mathbf{ A})^2\) in \(L^2(\Omega_\varepsilon)\). \( \mathbf{ A}: {\mathbb R}^2 \to {\mathbb R}^2\) be a vector field satisfying \( \mathrm{ curl}\ \mathbf{ A} = 0\) in \({\mathbb R}^2 \backslash K\), where \(K \subset \Omega\) is a compact set. Dirichlet boundary conditions are posed on \(C\) and Neumann boundary condition are posed on \(\{x: \mathrm{ dist}(x,C) = \varepsilon\}\). Denote by \((\lambda_n(\varepsilon)_{n\geq 1}\) the non-decreasing sequence of eigenvalues of \( \mathcal{L}_\varepsilon\). The authors derive the leading order term for the spectral gap \(\lambda_2(\varepsilon) - \lambda_1(\varepsilon)\) in the limit \(\varepsilon \to 0\).