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

Cornean, Horia D.; Fournais, Søren; Frank, Rupert L.; Helffer, Bernard

doi:10.5802/aif.2835

Sharp trace asymptotics for a class of

2 D

-magnetic operators
[Asymptotiques précisées pour la trace d’une classe d’opérateurs de Schrödinger magnétiques en dimension 2]

Cornean, Horia D.¹ ; Fournais, Søren ² ; Frank, Rupert L.³ ; Helffer, Bernard 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, Tome 63 (2013) no. 6, pp. 2457-2513.

Résumé
Abstract

Dans cet article, nous démontrons une formule asymptotique à deux termes pour la fonction de comptage spectrale de la réalisation de Dirichlet d’un opérateur de Schrödinger magnétique dans un domaine $Ω$ de $ℝ^{2}$ , en se plaçant dans la limite semi-classique et champ magnétique fort. Après changement d’échelle, ce problème est équivalent à celui de la limite thermodynamique pour un gaz de Fermi soumis à un champ magnétique extérieur constant. Notre motivation initiale provient d’un article de H. Kunz qui analyse entre autres choses l’influence de la frontière dans l’asymptotique de la pression et de la densité d’un tel gaz. Notre théorème donne une preuve rigoureuse des formules annoncées par Kunz et permet d’obtenir d’autres résultats pour des opérateurs du type ${(- i h \nabla - μ A)}^{2}$ dans $L^{2} (Ω)$ avec des conditions de Dirichlet au bord.

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 ${(- i h \nabla - μ A)}^{2}$ in $L^{2} (Ω)$ with Dirichlet boundary conditions.

MR Zbl

DOI : 10.5802/aif.2835

Classification : 35P20, 81V10
Keywords: Semiclassical asymptotics, Weyl law, magnetic Schrödinger operators
Mot clés : Asymptotique semiclassique, asymptotique de Weyl, opérateurs de Schrödinger avec champ magnétique

Affiliations des auteurs :

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

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     title = {Sharp trace asymptotics for a class of $2D$-magnetic operators},
     journal = {Annales de l'Institut Fourier},
     pages = {2457--2513},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Cornean, Horia D.; Fournais, Søren; Frank, Rupert L.; Helffer, Bernard. Sharp trace asymptotics for a class of $2D$-magnetic operators. Annales de l'Institut Fourier, Tome 63 (2013) no. 6, pp. 2457-2513. doi : 10.5802/aif.2835. http://www.numdam.org/articles/10.5802/aif.2835/

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