Eigenvalue estimates for a three-dimensional magnetic Schr "odinger\n operator

2012/03/18 by Bernard Helffer, Helffer, Bernard, Yuri A. Kordyukov +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1203.4021

openalex publication_date 2012/03/18 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We consider a magnetic Schr "odinger operator H^h=(-ih\∇-\A)²\nwith the Dirichlet boundary conditions in an open set \Ω \⊂ mathbb\nR³, where h>0 is a small parameter. We suppose that the minimal value\nb₀ of the module |\B| of the vector magnetic field \B is\nstrictly positive, and there exists a unique minimum point of |\B|,\nwhich is non-degenerate. The main result of the paper is upper estimates for\nthe low-lying eigenvalues of the operator H^h in the semiclassical limit. We\nalso prove the existence of an arbitrary large number of spectral gaps in the\nsemiclassical limit in the corresponding periodic setting.\n

Related

• Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator: The case of discrete wells
• On eigenvalue accumulation for non-self-adjoint magnetic operators
• Semiclassical spectral asymptotics for a magnetic Schrödinger operator with non-vanishing magnetic field
• Absence of positive eigenvalues of magnetic Schrödinger operators
• Eigenvalue asymptotics for confining magnetic Schrödinger operators with complex potentials