# Stability and semiclassics in self-generated fields

László Erdős; Soren Fournais; Jan Philip Solovej

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 6, page 2093-2113
- ISSN: 1435-9855

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topErdős, László, Fournais, Soren, and Solovej, Jan Philip. "Stability and semiclassics in self-generated fields." Journal of the European Mathematical Society 015.6 (2013): 2093-2113. <http://eudml.org/doc/277636>.

@article{Erdős2013,

abstract = {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 $\beta \int B^2$ 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 $\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, $h\rightarrow 0$, of the total ground state energy $E(\beta , h, V)$. The relevant parameter measuring the field strength in the semiclassical limit is $\kappa =\beta h$. We are not able to give the exact leading order semiclassical asymptotics uniformly in $\kappa $ or even for fixed $\kappa $. We do however give upper and lower bounds on $E$ with almost matching dependence on $\kappa $. In the simultaneous limit $h\rightarrow 0$ and $\kappa \rightarrow \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.},

author = {Erdős, László, Fournais, Soren, Solovej, Jan Philip},

journal = {Journal of the European Mathematical Society},

keywords = {semiclassical eigenvalue estimate; Maxwell-Pauli system; Scott correction; semiclassical eigenvalue estimate; Maxwell-Pauli system; Scott correction},

language = {eng},

number = {6},

pages = {2093-2113},

publisher = {European Mathematical Society Publishing House},

title = {Stability and semiclassics in self-generated fields},

url = {http://eudml.org/doc/277636},

volume = {015},

year = {2013},

}

TY - JOUR

AU - Erdős, László

AU - Fournais, Soren

AU - Solovej, Jan Philip

TI - Stability and semiclassics in self-generated fields

JO - Journal of the European Mathematical Society

PY - 2013

PB - European Mathematical Society Publishing House

VL - 015

IS - 6

SP - 2093

EP - 2113

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 $\beta \int B^2$ 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 $\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, $h\rightarrow 0$, of the total ground state energy $E(\beta , h, V)$. The relevant parameter measuring the field strength in the semiclassical limit is $\kappa =\beta h$. We are not able to give the exact leading order semiclassical asymptotics uniformly in $\kappa $ or even for fixed $\kappa $. We do however give upper and lower bounds on $E$ with almost matching dependence on $\kappa $. In the simultaneous limit $h\rightarrow 0$ and $\kappa \rightarrow \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.

LA - eng

KW - semiclassical eigenvalue estimate; Maxwell-Pauli system; Scott correction; semiclassical eigenvalue estimate; Maxwell-Pauli system; Scott correction

UR - http://eudml.org/doc/277636

ER -

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