Stability and semiclassics in self-generated fields

TY - JOUR

T1 - Stability and semiclassics in self-generated fields

AU - Erdös, Laszlo

AU - Fournais, Søren

AU - Solovej, Jan Philip

PY - 2013

Y1 - 2013

N2 - 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 f 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 k = βh. We are not able to give the exact leading order semiclassical asymptotics uniformly in k or even for fixed k. We do however give upper and lower bounds on E with almost matching dependence on k. In the simultaneous limit h → 0 and k → ∞ 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.

AB - 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 f 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 k = βh. We are not able to give the exact leading order semiclassical asymptotics uniformly in k or even for fixed k. We do however give upper and lower bounds on E with almost matching dependence on k. In the simultaneous limit h → 0 and k → ∞ 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.

M3 - Journal article

SN - 1435-9855

VL - 15

SP - 2093

EP - 2113

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 6

ER -