Sharp trace asymptotics for a class of 2 D -magnetic operators

Horia D. Cornean[1]; Søren Fournais[2]; Rupert L. Frank[3]; Bernard Helffer[4]

  • [1] Aalborg University Department of Mathematical Sciences Fredrik Bajers Vej 7G 9220 Aalborg (Denmark)
  • [2] Aarhus University Department of Mathematics Ny Munkegade 181, Building 1530 8000 Aarhus C (Denmark)
  • [3] Princeton University Fine Hall Department of Mathematics Princeton, NJ 08544 (USA)
  • [4] Université Paris Sud et CNRS Laboratoire de Mathématiques Bâtiment 425 91405 Orsay Cedex (France)

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 6, page 2457-2513
  • ISSN: 0373-0956

Abstract

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In this paper we prove a two-term asymptotic formula for the spectral counting function for a 2 D magnetic Schrödinger operator on a domain (with Dirichlet boundary conditions) in a semiclassical limit and with strong magnetic field. By scaling, this is equivalent to a thermodynamic limit of a 2 D Fermi gas submitted to a constant external magnetic field.The original motivation comes from a paper by H. Kunz in which he studied, among other things, the boundary correction for the grand-canonical pressure and density of such a Fermi gas. Our main theorem yields a rigorous proof of the formulas announced by Kunz. Moreover, the same theorem provides several other results on the integrated density of states for operators of the type ( - i h ∇ - μ A ) 2 in L 2 ( Ω ) with Dirichlet boundary conditions.

How to cite

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Cornean, Horia D., et al. "Sharp trace asymptotics for a class of $2D$-magnetic operators." Annales de l’institut Fourier 63.6 (2013): 2457-2513. <http://eudml.org/doc/275472>.

@article{Cornean2013,
abstract = {In this paper we prove a two-term asymptotic formula for the spectral counting function for a $2$D magnetic Schrödinger operator on a domain (with Dirichlet boundary conditions) in a semiclassical limit and with strong magnetic field. By scaling, this is equivalent to a thermodynamic limit of a $2$D Fermi gas submitted to a constant external magnetic field.The original motivation comes from a paper by H. Kunz in which he studied, among other things, the boundary correction for the grand-canonical pressure and density of such a Fermi gas. Our main theorem yields a rigorous proof of the formulas announced by Kunz. Moreover, the same theorem provides several other results on the integrated density of states for operators of the type $(-ih\nabla - \mu \{\bf A\})^2$ in $L^2(\{\Omega \})$ with Dirichlet boundary conditions.},
affiliation = {Aalborg University Department of Mathematical Sciences Fredrik Bajers Vej 7G 9220 Aalborg (Denmark); Aarhus University Department of Mathematics Ny Munkegade 181, Building 1530 8000 Aarhus C (Denmark); Princeton University Fine Hall Department of Mathematics Princeton, NJ 08544 (USA); Université Paris Sud et CNRS Laboratoire de Mathématiques Bâtiment 425 91405 Orsay Cedex (France)},
author = {Cornean, Horia D., Fournais, Søren, Frank, Rupert L., Helffer, Bernard},
journal = {Annales de l’institut Fourier},
keywords = {Semiclassical asymptotics; Weyl law; magnetic Schrödinger operators; semiclassical asymptotics},
language = {eng},
number = {6},
pages = {2457-2513},
publisher = {Association des Annales de l’institut Fourier},
title = {Sharp trace asymptotics for a class of $2D$-magnetic operators},
url = {http://eudml.org/doc/275472},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Cornean, Horia D.
AU - Fournais, Søren
AU - Frank, Rupert L.
AU - Helffer, Bernard
TI - Sharp trace asymptotics for a class of $2D$-magnetic operators
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 6
SP - 2457
EP - 2513
AB - In this paper we prove a two-term asymptotic formula for the spectral counting function for a $2$D magnetic Schrödinger operator on a domain (with Dirichlet boundary conditions) in a semiclassical limit and with strong magnetic field. By scaling, this is equivalent to a thermodynamic limit of a $2$D Fermi gas submitted to a constant external magnetic field.The original motivation comes from a paper by H. Kunz in which he studied, among other things, the boundary correction for the grand-canonical pressure and density of such a Fermi gas. Our main theorem yields a rigorous proof of the formulas announced by Kunz. Moreover, the same theorem provides several other results on the integrated density of states for operators of the type $(-ih\nabla - \mu {\bf A})^2$ in $L^2({\Omega })$ with Dirichlet boundary conditions.
LA - eng
KW - Semiclassical asymptotics; Weyl law; magnetic Schrödinger operators; semiclassical asymptotics
UR - http://eudml.org/doc/275472
ER -

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