Sharp trace asymptotics for a class of 2D-magnetic operators
[Asymptotiques précisées pour la trace d’une classe d’opérateurs de Schrödinger magnétiques en dimension 2]
Annales de l'Institut Fourier, Tome 63 (2013) no. 6, pp. 2457-2513.

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 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 (-ih-μA) 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 2D 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 2D 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-μA) 2 in L 2 (Ω) with Dirichlet boundary conditions.

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
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)
@article{AIF_2013__63_6_2457_0,
     author = {Cornean, Horia D. and Fournais, S{\o}ren and Frank, Rupert L. and Helffer, Bernard},
     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},
     volume = {63},
     number = {6},
     year = {2013},
     doi = {10.5802/aif.2835},
     mrnumber = {3237453},
     zbl = {1301.35070},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2835/}
}
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
SP  - 2457
EP  - 2513
VL  - 63
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2835/
DO  - 10.5802/aif.2835
LA  - en
ID  - AIF_2013__63_6_2457_0
ER  - 
%0 Journal Article
%A Cornean, Horia D.
%A Fournais, Søren
%A Frank, Rupert L.
%A Helffer, Bernard
%T Sharp trace asymptotics for a class of $2D$-magnetic operators
%J Annales de l'Institut Fourier
%D 2013
%P 2457-2513
%V 63
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2835/
%R 10.5802/aif.2835
%G en
%F AIF_2013__63_6_2457_0
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. https://aif.centre-mersenne.org/articles/10.5802/aif.2835/

[1] Briet, Philippe; Hislop, Peter D.; Raikov, Georgi; Soccorsi, Eric Mourre estimates for a 2D magnetic quantum Hamiltonian on strip-like domains, Spectral and scattering theory for quantum magnetic systems (Contemp. Math.), Volume 500, Amer. Math. Soc., Providence, RI, 2009, pp. 33-46 | DOI | MR | Zbl

[2] Briet, Philippe; Raikov, Georgi; Soccorsi, Eric Spectral properties of a magnetic quantum Hamiltonian on a strip, Asymptot. Anal., Volume 58 (2008) no. 3, pp. 127-155 | MR | Zbl

[3] Broderix, Kurt; Hundertmark, Dirk; Leschke, Hajo Continuity properties of Schrödinger semigroups with magnetic fields, Rev. Math. Phys., Volume 12 (2000) no. 2, pp. 181-225 | DOI | MR | Zbl

[4] Combes, J. M.; Thomas, L. Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators, Comm. Math. Phys., Volume 34 (1973), pp. 251-270 | DOI | MR | Zbl

[5] Cornean, H. D. On spectral properties of Dirac or Schrödinger operators with magnetic field, Bucarest (1999) (Ph. D. Thesis)

[6] Cornean, H. D.; Nenciu, Gheorghe The Faraday effect revisited: thermodynamic limit, J. Funct. Anal., Volume 257 (2009) no. 7, pp. 2024-2066 | DOI | MR | Zbl

[7] De Bièvre, Stephan; Pulé, Joseph V. Propagating edge states for a magnetic Hamiltonian, Math. Phys. Electron. J., Volume 5 (1999), pp. Paper 3, 17 pp. (electronic) | MR | Zbl

[8] Dimassi, Mouez; Sjöstrand, Johannes Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268, Cambridge University Press, Cambridge, 1999, pp. xii+227 | DOI | MR | Zbl

[9] Erdős, László; Solovej, Jan Philip Semiclassical eigenvalue estimates for the Pauli operator with strong non-homogeneous magnetic fields. II. Leading order asymptotic estimates, Comm. Math. Phys., Volume 188 (1997) no. 3, pp. 599-656 | DOI | MR | Zbl

[10] Fournais, Søren; Helffer, Bernard Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian, Ann. Inst. Fourier (Grenoble), Volume 56 (2006) no. 1, pp. 1-67 | DOI | Numdam | MR | Zbl

[11] Fournais, Søren; Helffer, Bernard Spectral methods in surface superconductivity, Progress in Nonlinear Differential Equations and their Applications, 77, Birkhäuser Boston Inc., Boston, MA, 2010, pp. xx+324 | MR | Zbl

[12] Fournais, Søren; Kachmar, Ayman On the energy of bound states for magnetic Schrödinger operators, J. Lond. Math. Soc. (2), Volume 80 (2009) no. 1, pp. 233-255 | DOI | MR | Zbl

[13] Frank, Rupert L. On the asymptotic number of edge states for magnetic Schrödinger operators, Proc. Lond. Math. Soc. (3), Volume 95 (2007) no. 1, pp. 1-19 | DOI | MR | Zbl

[14] Frank, Rupert L.; Loss, Michael; Weidl, Timo Pólya’s conjecture in the presence of a constant magnetic field, J. Eur. Math. Soc. (JEMS), Volume 11 (2009) no. 6, pp. 1365-1383 | DOI | MR | Zbl

[15] Ghribi, Fatma Internal Lifshits tails for random magnetic Schrödinger operators, J. Funct. Anal., Volume 248 (2007) no. 2, pp. 387-427 | DOI | MR | Zbl

[16] Helffer, B.; Sjöstrand, J. On diamagnetism and de Haas-van Alphen effect, Ann. Inst. H. Poincaré Phys. Théor., Volume 52 (1990) no. 4, pp. 303-375 | Numdam | MR | Zbl

[17] Helffer, Bernard Semi-classical analysis for the Schrödinger operator and applications, Lecture Notes in Mathematics, 1336, Springer-Verlag, Berlin, 1988, pp. vi+107 | MR | Zbl

[18] Helffer, Bernard; Mohamed, Abderemane Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells, J. Funct. Anal., Volume 138 (1996) no. 1, pp. 40-81 | DOI | MR | Zbl

[19] Helffer, Bernard; Morame, Abderemane Magnetic bottles in connection with superconductivity, J. Funct. Anal., Volume 185 (2001) no. 2, pp. 604-680 | DOI | MR | Zbl

[20] Helffer, Bernard; Robert, D. Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles, J. Funct. Anal., Volume 53 (1983) no. 3, pp. 246-268 | DOI | MR | Zbl

[21] Helffer, Bernard; Sjöstrand, J. Équation de Schrödinger avec champ magnétique et équation de Harper, Schrödinger operators (Sønderborg, 1988) (Lecture Notes in Phys.), Volume 345, Springer, Berlin, 1989, pp. 118-197 | DOI | MR | Zbl

[22] Hornberger, Klaus; Smilansky, Uzy Magnetic edge states, Phys. Rep., Volume 367 (2002) no. 4, pp. 249-385 | DOI | MR

[23] Hupfer, Thomas; Leschke, Hajo; Müller, Peter; Warzel, Simone Existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials, Rev. Math. Phys., Volume 13 (2001) no. 12, pp. 1547-1581 | DOI | MR | Zbl

[24] Ivriĭ, V. Ja. The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary, Funktsional. Anal. i Prilozhen., Volume 14 (1980) no. 2, pp. 25-34 | DOI | MR | Zbl

[25] Kunz, Hervé Surface orbital magnetism, J. Statist. Phys., Volume 76 (1994) no. 1-2, pp. 183-207 | DOI | MR | Zbl

[26] Lieb, Elliott H.; Solovej, Jan Philip; Yngvason, Jakob Asymptotics of heavy atoms in high magnetic fields. II. Semiclassical regions, Comm. Math. Phys., Volume 161 (1994) no. 1, pp. 77-124 http://projecteuclid.org/getRecord?id=euclid.cmp/1104269793 | DOI | MR | Zbl

[27] Persson, Mikael Eigenvalue asymptotics of the even-dimensional exterior Landau-Neumann Hamiltonian, Adv. Math. Phys. (2009), pp. Art. ID 873704, 15 | DOI | MR | Zbl

[28] Pushnitski, Alexander; Rozenblum, Grigori Eigenvalue clusters of the Landau Hamiltonian in the exterior of a compact domain, Doc. Math., Volume 12 (2007), pp. 569-586 | MR | Zbl

[29] Safarov, Yu.; Vassiliev, D. The asymptotic distribution of eigenvalues of partial differential operators, Translations of Mathematical Monographs, 155, American Mathematical Society, Providence, RI, 1997, pp. xiv+354 (Translated from the Russian manuscript by the authors) | MR | Zbl

[30] Sobolev, A. V. The quasi-classical asymptotics of local Riesz means for the Schrödinger operator in a strong homogeneous magnetic field, Duke Math. J., Volume 74 (1994) no. 2, pp. 319-429 | DOI | MR | Zbl

[31] Sobolev, A. V. Quasi-classical asymptotics of local Riesz means for the Schrödinger operator in a moderate magnetic field, Ann. Inst. H. Poincaré Phys. Théor., Volume 62 (1995) no. 4, pp. 325-360 | Numdam | MR | Zbl

[32] Sobolev, A. V. Quasi-classical asymptotics for the Pauli operator, Comm. Math. Phys., Volume 194 (1998) no. 1, pp. 109-134 | DOI | MR | Zbl

[33] Tamura, Hideo Asymptotic distribution of eigenvalues for Schrödinger operators with magnetic fields, Nagoya Math. J., Volume 105 (1987), pp. 49-69 http://projecteuclid.org/getRecord?id=euclid.nmj/1118780638 | MR | Zbl

[34] Colin de Verdière, Yves L’asymptotique de Weyl pour les bouteilles magnétiques, Comm. Math. Phys., Volume 105 (1986) no. 2, pp. 327-335 http://projecteuclid.org/getRecord?id=euclid.cmp/1104115337 | DOI | MR | Zbl

Cité par Sources :