JournalsjemsVol. 15, No. 6pp. 2093–2113

Stability and semiclassics in self-generated fields

  • Søren Fournais

    University of Aarhus, Denmark
  • Jan Philip Solovej

    University of Copenhagen, Denmark
  • László Erdős

    Institute of Scienceand Technology Austria, Klosterneuburg, Austria
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Abstract

We consider non-interacting particles subject to a fixed external potential VV and a self-generated magnetic field BB. The total energy includes the field energy βB2\beta \int B^2 and we minimize over all particle states and magnetic fields. In the case of spin-1/21/2 particles this minimization leads to the coupled Maxwell-Pauli system. The parameter β\beta 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, h0h\to0, of the total ground state energy E(β,h,V)E(\beta, h, V). The relevant parameter measuring the field strength in the semiclassical limit is κ=βh\kappa=\beta h. We are not able to give the exact leading order semiclassical asymptotics uniformly in κ\kappa or even for fixed κ\kappa. We do however give upper and lower bounds on EE with almost matching dependence on κ\kappa. In the simultaneous limit h0h\to0 and κ\kappa\to\infty 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.

Cite this article

Søren Fournais, Jan Philip Solovej, László Erdős, Stability and semiclassics in self-generated fields. J. Eur. Math. Soc. 15 (2013), no. 6, pp. 2093–2113

DOI 10.4171/JEMS/416