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]
Cornean, Horia D. ; Fournais, Søren ; Frank, Rupert L. ; Helffer, Bernard
Annales de l'Institut Fourier, Tome 63 (2013), p. 2457-2513 / Harvested from Numdam
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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 (-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.

Publié le : 2013-01-01
DOI : https://doi.org/10.5802/aif.2835
Classification:  35P20,  81V10
Mots clés: Asymptotique semiclassique, asymptotique de Weyl, opérateurs de Schrödinger avec champ magnétique 
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     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},
     volume = {63},
     year = {2013},
     pages = {2457-2513},
     doi = {10.5802/aif.2835},
     zbl = {1301.35070},
     mrnumber = {3237453},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2013__63_6_2457_0}
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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) pp. 2457-2513. doi : 10.5802/aif.2835. http://gdmltest.u-ga.fr/item/AIF_2013__63_6_2457_0/

[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, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 500 (2009), pp. 33-46 | Article | MR 2655141 | Zbl 1183.81063

[2] Briet, Philippe; Raikov, Georgi; Soccorsi, Eric Spectral properties of a magnetic quantum Hamiltonian on a strip, Asymptot. Anal., Tome 58 (2008) no. 3, pp. 127-155 | MR 2456460 | Zbl 1163.35446

[3] Broderix, Kurt; Hundertmark, Dirk; Leschke, Hajo Continuity properties of Schrödinger semigroups with magnetic fields, Rev. Math. Phys., Tome 12 (2000) no. 2, pp. 181-225 | Article | MR 1756112 | Zbl 0961.81006

[4] Combes, J. M.; Thomas, L. Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators, Comm. Math. Phys., Tome 34 (1973), pp. 251-270 | Article | MR 391792 | Zbl 0271.35062

[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., Tome 257 (2009) no. 7, pp. 2024-2066 | Article | MR 2548029 | Zbl 1178.82077

[7] De Bièvre, Stephan; Pulé, Joseph V. Propagating edge states for a magnetic Hamiltonian, Math. Phys. Electron. J., Tome 5 (1999), pp. Paper 3, 17 pp. (electronic) | MR 1703586 | Zbl 0930.35144

[8] Dimassi, Mouez; Sjöstrand, Johannes Spectral asymptotics in the semi-classical limit, Cambridge University Press, Cambridge, London Mathematical Society Lecture Note Series, Tome 268 (1999), pp. xii+227 | Article | MR 1735654 | Zbl 0926.35002

[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., Tome 188 (1997) no. 3, pp. 599-656 | Article | MR 1473314 | Zbl 0909.47052

[10] Fournais, Søren; Helffer, Bernard Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian, Ann. Inst. Fourier (Grenoble), Tome 56 (2006) no. 1, pp. 1-67 http://aif.cedram.org/item?id=AIF_2006__56_1_1_0 | Article | Numdam | MR 2228679 | Zbl 1097.47020

[11] Fournais, Søren; Helffer, Bernard Spectral methods in surface superconductivity, Birkhäuser Boston Inc., Boston, MA, Progress in Nonlinear Differential Equations and their Applications, 77 (2010), pp. xx+324 | MR 2662319 | Zbl 1256.35001

[12] Fournais, Søren; Kachmar, Ayman On the energy of bound states for magnetic Schrödinger operators, J. Lond. Math. Soc. (2), Tome 80 (2009) no. 1, pp. 233-255 | Article | MR 2520387 | Zbl 1179.35203

[13] Frank, Rupert L. On the asymptotic number of edge states for magnetic Schrödinger operators, Proc. Lond. Math. Soc. (3), Tome 95 (2007) no. 1, pp. 1-19 | Article | MR 2329546 | Zbl 1131.35076

[14] Frank, Rupert L.; Loss, Michael; Weidl, Timo Pólya’s conjecture in the presence of a constant magnetic field, J. Eur. Math. Soc. (JEMS), Tome 11 (2009) no. 6, pp. 1365-1383 | Article | MR 2557138 | Zbl 1179.35205

[15] Ghribi, Fatma Internal Lifshits tails for random magnetic Schrödinger operators, J. Funct. Anal., Tome 248 (2007) no. 2, pp. 387-427 | Article | MR 2335580 | Zbl 1121.82022

[16] Helffer, B.; Sjöstrand, J. On diamagnetism and de Haas-van Alphen effect, Ann. Inst. H. Poincaré Phys. Théor., Tome 52 (1990) no. 4, pp. 303-375 | Numdam | MR 1062904 | Zbl 0715.35070

[17] Helffer, Bernard Semi-classical analysis for the Schrödinger operator and applications, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1336 (1988), pp. vi+107 | MR 960278 | Zbl 0647.35002

[18] Helffer, Bernard; Mohamed, Abderemane Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells, J. Funct. Anal., Tome 138 (1996) no. 1, pp. 40-81 | Article | MR 1391630 | Zbl 0851.58046

[19] Helffer, Bernard; Morame, Abderemane Magnetic bottles in connection with superconductivity, J. Funct. Anal., Tome 185 (2001) no. 2, pp. 604-680 | Article | MR 1856278 | Zbl 1078.81023

[20] Helffer, Bernard; Robert, D. Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles, J. Funct. Anal., Tome 53 (1983) no. 3, pp. 246-268 | Article | MR 724029 | Zbl 0524.35103

[21] Helffer, Bernard; Sjöstrand, J. Équation de Schrödinger avec champ magnétique et équation de Harper, Schrödinger operators (Sønderborg, 1988), Springer, Berlin (Lecture Notes in Phys.) Tome 345 (1989), pp. 118-197 | Article | MR 1037319 | Zbl 0699.35189

[22] Hornberger, Klaus; Smilansky, Uzy Magnetic edge states, Phys. Rep., Tome 367 (2002) no. 4, pp. 249-385 | Article | MR 1921565

[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., Tome 13 (2001) no. 12, pp. 1547-1581 | Article | MR 1869817 | Zbl 1029.81027

[24] Ivriĭ, V. Ja. The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary, Funktsional. Anal. i Prilozhen., Tome 14 (1980) no. 2, pp. 25-34 | Article | MR 575202 | Zbl 0453.35068

[25] Kunz, Hervé Surface orbital magnetism, J. Statist. Phys., Tome 76 (1994) no. 1-2, pp. 183-207 | Article | MR 1297876 | Zbl 1080.82506

[26] Lieb, Elliott H.; Solovej, Jan Philip; Yngvason, Jakob Asymptotics of heavy atoms in high magnetic fields. II. Semiclassical regions, Comm. Math. Phys., Tome 161 (1994) no. 1, pp. 77-124 http://projecteuclid.org/getRecord?id=euclid.cmp/1104269793 | Article | MR 1266071 | Zbl 0807.47058

[27] Persson, Mikael Eigenvalue asymptotics of the even-dimensional exterior Landau-Neumann Hamiltonian, Adv. Math. Phys. (2009), pp. Art. ID 873704, 15 | Article | MR 2500946 | Zbl 1201.81055

[28] Pushnitski, Alexander; Rozenblum, Grigori Eigenvalue clusters of the Landau Hamiltonian in the exterior of a compact domain, Doc. Math., Tome 12 (2007), pp. 569-586 | MR 2377242 | Zbl 1132.35424

[29] Safarov, Yu.; Vassiliev, D. The asymptotic distribution of eigenvalues of partial differential operators, American Mathematical Society, Providence, RI, Translations of Mathematical Monographs, Tome 155 (1997), pp. xiv+354 (Translated from the Russian manuscript by the authors) | MR 1414899 | Zbl 0898.35003

[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., Tome 74 (1994) no. 2, pp. 319-429 | Article | MR 1272980 | Zbl 0824.35151

[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., Tome 62 (1995) no. 4, pp. 325-360 | Numdam | MR 1343781 | Zbl 0843.35024

[32] Sobolev, A. V. Quasi-classical asymptotics for the Pauli operator, Comm. Math. Phys., Tome 194 (1998) no. 1, pp. 109-134 | Article | MR 1628306 | Zbl 0915.47052

[33] Tamura, Hideo Asymptotic distribution of eigenvalues for Schrödinger operators with magnetic fields, Nagoya Math. J., Tome 105 (1987), pp. 49-69 http://projecteuclid.org/getRecord?id=euclid.nmj/1118780638 | MR 881008 | Zbl 0623.35048

[34] Colin De Verdière, Yves L’asymptotique de Weyl pour les bouteilles magnétiques, Comm. Math. Phys., Tome 105 (1986) no. 2, pp. 327-335 http://projecteuclid.org/getRecord?id=euclid.cmp/1104115337 | Article | MR 849211 | Zbl 0612.35102