Semiclassical analysis with vanishing magnetic fields (Q361728)

A complete asymptotic expansion of the eigenvalues is obtained for the magnetic two-dimensional Hamiltonian \((-ih\nabla +A)^{2}\) for a vanishing magnetic field \(\beta = \nabla \times A\) along a closed smooth curve. More precisely, the authors prove that if the vector potential \(A\) is smooth, \(\beta(x) \rightarrow \infty\) when \(| x | \rightarrow \infty\), and if the normal derivative of \(\beta\) on the curve admits a unique, nondegenerate and positive minimum, then for all \(n \geq 1\) there exists a sequence \((\theta^{n}_{j})_{j\geq 0}\) and \(h_{0} \geq 0\) such that the \(n\)-th eigenvalue \(\lambda _{n}(h) \sim h^{4/3} \sum_{j\geq 0} \theta^{n}_{j} h^{j/6}\). Exact expressions for \(\theta^{n}_{0}\), \(\theta^{n}_{1}\) and \(\theta^{n}_{2}\) are given. In the process of the proof a spectral gap between the eigenvalues is also obtained.