Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator: The case of discrete wells

Abstract

Abstract. We consider a magnetic Schrödinger operator H h , depending on the semiclassical parameter h > 0, on a two-dimensional Riemannian manifold. We assume that there is no electric field. We suppose that the minimal value b 0 of the magnetic field b is strictly positive, and there exists a unique minimum point of b, which is non-degenerate. The main result of the paper is a complete asymptotic expansion for the low-lying eigenvalues of the operator H h in the semiclassical limit. We also apply these results to prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting.

When the 3D Magnetic Laplacian Meets a Curved Edge in the Semiclassical Limit

“…In our paper we consider a case when the boundary is not smooth: the case when ∂Ω contains an edge. Our analysis will show that, even in the non smooth case, the repartition of the low lying eigenvalues is determined by an effective 1D harmonic oscillator with respect to the Fourier variable on the edge (see [16,22,37,36,14]). …”

The Ginzburg–Landau Functional with Vanishing Magnetic Field

“…Under the assumption (2.12), the known spectral asymptotics (which are actually the same in this case) of the Dirichlet and Neumann eigenvalues will lead us to the asymptotics given in the righthand side of (2.13). Under the additional assumption that inf x∈Ω |B 0 (x)| is attained at a unique minimum in Ω and that this minimum is non degenerate, a complete asymptotics of λ N (tA 0 ) can be given (see Helffer-Mohamed [15], Helffer-Kordyukov [13,14], Raymond-Vu Ngoc [23] ) and the monotonicity/strong diamagnetism property holds for large values of t (see Chapter 3 in [4]). Hence the definition of H c 3 (κ) is clear in this case.…”

Microlocal Normal Forms for the Magnetic Laplacian

“…recent works by Helffer-Kordyukov [6]). Indeed, by specializing to the case where B admits a non-degenerate minimum, we obtain a full asymptotic expansions of low eigenvalues in integer powers of : Corollary 3.5.…”