2015/09/24 by Denis Bonheure, Silvia Cingolani, Bonheure, Denis +3 · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics #math.AP
paper · pdf · doi:10.48550/arxiv.1509.07464
arxiv created 2015/09/24 · openalex publication_date 2015/09/24 · arxiv updated 2015/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the semiclassical limit for the stationary magnetic nonlinear Schrödinger equation ( i ℏ ∇ + A(x) )2 u + V(x) u = |u|p-2 u, x∈ ℝ3,where p\textgreater2, A is a vector potential associated to a given magnetic field B, i.e ∇ × A =B and V is a nonnegative, scalar (electric) potential which can be singular at the origin and vanish at infinity or outside a compact set.We assume that A and V satisfy a cylindrical symmetry. By a refined penalization argument, we prove the existence of semiclassical cylindrically symmetric solutions of upper equation whose moduli concentrate, as ℏ → 0, around a circle. We emphasize that the concentration is driven by the magnetic and the electric potentials. Our result thus shows that in the semiclassical limit, the magnetic field also influences the location of the solutions of (\refeq:initialabstract) if their concentration occurs around a locus, not a single point.