2011/05/03 by Laszlo Erdos, Erdos, Laszlo, Soren Fournais +3
Mathematics · Physics and Astronomy · #35P15 #81Q10 #81Q20 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.MP #math.SP #msc:35P15 #msc:81Q10 #msc:81Q20
paper · pdf · doi:10.48550/arxiv.1105.0506
New version of October 18 with substantial changes compared to previous version. Conjectures have been replaced by Theorems
arxiv created 2011/10/20 · arxiv updated 2011/10/21
We consider non-interacting particles subject to a fixed external potential V and a self-generated magnetic field B. The total energy includes the field energy β∫ B2 and we minimize over all particle states and magnetic fields. In the case of spin-1/2 particles this minimization leads to the coupled Maxwell-Pauli system. The parameter β tunes the coupling strength between the field and the particles and it effectively determines the strength of the field. We investigate the stability and the semiclassical asymptotics, h→0, of the total ground state energy E(β, h, V). The relevant parameter measuring the field strength in the semiclassical limit is κ=βh. We are not able to give the exact leading order semiclassical asymptotics uniformly in κ or even for fixed κ. We do however give upper and lower bounds on E with almost matching dependence on κ. In the simultaneous limit h→0 and κ→∞ we show that the standard non-magnetic Weyl asymptotics holds. The same result also holds for the spinless case, i.e. where the Pauli operator is replaced by the Schrödinger operator.