Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian  [ Estimations asymptotiques précises pour le laplacien magnétique de Neumann ]
Annales de l'Institut Fourier, Tome 56 (2006) no. 1, p. 1-67
Motivés par la théorie de la supraconductivité et plus précisément par le problème de l’apparition de la supraconductivité à la surface, de nombreux articles ont été consacrés récemment à l’analyse semi-classique de la plus petite valeur propre de l’opérateur de Schrödinger avec champ magnétique (Bernoff-Sternberg, Lu-Pan, Del Pino-Felmer-Sternberg, Helffer-Morame et aussi Bauman-Phillips-Tang pour le cas du disque). Dans cet article, nous proposons des asymptotiques complètes pour les premières valeurs propres dans le cas d’un domaine de 2 dont la courbure du bord n’a qu’un unique maximum non-dégénéré.
Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, many papers devoted to the analysis in a semi-classical regime of the lowest eigenvalue of the Schrödinger operator with magnetic field have appeared recently. Here we would like to mention the works by Bernoff-Sternberg, Lu-Pan, Del Pino-Felmer-Sternberg and Helffer-Morame and also Bauman-Phillips-Tang for the case of a disc. In the present paper we settle one important part of this question completely by proving an asymptotic expansion to all orders for low-lying eigenvalues for generic domains. The word ‘generic’ means in this context that the curvature of the boundary of the domain has a unique non-degenerate maximum.
DOI : https://doi.org/10.5802/aif.2171
Classification:  47A75,  58C40,  35Q40,  81Q20
Mots clés: analyse semi-classique, supraconductivité, laplacien de Neumann, laplacien magnétique
@article{AIF_2006__56_1_1_0,
     author = {Fournais, Soeren and Helffer, Bernard},
     title = {Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {1},
     year = {2006},
     pages = {1-67},
     doi = {10.5802/aif.2171},
     mrnumber = {2228679},
     zbl = {1097.47020},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2006__56_1_1_0}
}
Fournais, Soeren; Helffer, Bernard. Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian. Annales de l'Institut Fourier, Tome 56 (2006) no. 1, pp. 1-67. doi : 10.5802/aif.2171. http://www.numdam.org/item/AIF_2006__56_1_1_0/

[1] Agmon, S. Lectures on exponential decay of solutions of second order elliptic equations, Princeton University Press, Math. Notes, Tome 29 (1982) | MR 745286 | Zbl 0503.35001

[2] Bauman, P.; Phillips, D.; Tang, Q. Stable nucleation for the Ginzburg-Landau system with an applied magnetic field, Arch. Rational Mech. Anal., Tome 142 (1998), pp. 1-43 | Article | MR 1629119 | Zbl 0922.35157

[3] Bernoff, A.; Sternberg, P. Onset of superconductivity in decreasing fields for general domains, J. Math. Phys., Tome 39 (1998), pp. 1272-1284 | Article | MR 1608449 | Zbl 1056.82523

[4] Bolley, C.; Helffer, B. An application of semi-classical analysis to the asymptotic study of the supercooling field of a superconducting material, Ann. Inst. H. Poincaré (Section Physique Théorique), Tome 58 (1993) no. 2, pp. 169-233 | Numdam | MR 1217119 | Zbl 0779.35104

[5] Bonnaillie, V. Analyse mathématique de la supraconductivité dans un domaine à coins : méthodes semi-classiques et numériques, Université Paris 11 (2003) (Ph. D. Thesis)

[6] Bonnaillie, V. On the fundamental state for a Schrödinger operator with magnetic fields in domains with corners, Asymptotic Anal., Tome 41 (2005) no. 3-4, pp. 215-258 | MR 2127997 | Zbl 1067.35054

[7] Bonnaillie-Noël, V.; Dauge, M. Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners (2005) (preprint)

[8] Cycon, H.L.; Froese, R.G.; Kirsch, W.; Simon, B. Schrödinger Operators, Springer Verlag, Berlin (1987) | Zbl 0619.47005

[9] Dauge, M.; Helffer, B. Eigenvalues variation I, Neumann problem for Sturm-Liouville operators, J. Differential Equations, Tome 104 (1993) no. 2, pp. 243-262 | Article | MR 1231468 | Zbl 0784.34021

[10] Dimassi, M.; Sjöstrand, J. Spectral Asymptotics in the semi-classical limit, Cambridge University Press, London Mathematical Society Lecture Note Series, Tome 268 (1999) | MR 1735654 | Zbl 0926.35002

[11] Fournais, S.; Helffer, B. Energy asymptotics for type II superconductors (2004) (preprint) | MR 2174430 | Zbl 02227540

[12] Grušhin, V. V. Hypoelliptic differential equations and pseudodifferential operators with operator-valued symbols, Mat. Sb. (N.S.), Tome 88 (1972) no. 130, pp. 504-521 ((russian)) | Zbl 0255.35022

[13] Helffer, B. Introduction to the semiclassical analysis for the Schrödinger operator and applications, Springer Verlag, Lecture Notes in Math., Tome 1336 (1988) | Zbl 0647.35002

[14] Helffer, B.; Mohamed, A. 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

[15] Helffer, B.; Morame, A. Magnetic bottles in connection with superconductivity, J. Funct. Anal., Tome 185 (2001) no. 2, pp. 604-680 | Article | MR 1856278 | Zbl 01688735

[16] Helffer, B.; Morame, A. Magnetic bottles for the Neumann problem : curvature effect in the case of dimension 3 (General case), Ann. Sci. École Norm. Sup., Tome 37 (2004), pp. 105-170 | Numdam | MR 2050207 | Zbl 1057.35061

[17] Helffer, B.; Pan, X. Upper critical field and location of surface nucleation of superconductivity, Ann. Inst. H. Poincaré (Section Analyse non linéaire), Tome 20 (2003) no. 1, pp. 145-181 | Article | Numdam | MR 1958165 | Zbl 1060.35132

[18] Helffer, B.; Sjöstrand, J. Multiple wells in the semiclassical limit I, Comm. Partial Differential Equations, Tome 9 (1984) no. 4, pp. 337-408 | Article | MR 740094 | Zbl 0546.35053

[19] Helffer, B.; Sjöstrand, J. Puits multiples en limite semi-classique II – Interaction moléculaire – Symétries – Perturbations, Ann. Inst. H. Poincaré (Section Physique théorique), Tome 42 (1985) no. 2, pp. 127-212 | Numdam | MR 798695 | Zbl 0595.35031

[20] Helffer, B.; Sjöstrand, J. Puits multiples en limite semiclassique V – le cas des minipuits, Current topics in partial differential equations, Kinokuniya, Tokyo (1986), pp. 133-186 | Zbl 0628.35024

[21] Helffer, B.; Sjöstrand, J. Effet tunnel pour l’équation de Schrödinger avec champ magnétique, Ann. Scuola Norm. Sup. Pisa, Tome 14 (1987) no. 4, pp. 625-657 | Numdam | MR 963493 | Zbl 0699.35205

[22] Lu, K.; Pan, X-B. Eigenvalue problems of Ginzburg-Landau operator in bounded domains, J. Math. Phys., Tome 40 (1999) no. 6, pp. 2647-2670 | Article | MR 1694223 | Zbl 0943.35058

[23] Lu, K.; Pan, X-B. Estimates of the upper critical field for the equations of superconductivity, Physica D, Tome 127 (1999), pp. 73-104 | Article | MR 1678383 | Zbl 0934.35174

[24] Lu, K.; Pan, X-B. Gauge invariant eigenvalue problems on 2 and + 2 , Trans. Amer. Math. Soc., Tome 352 (2000) no. 3, pp. 1247-1276 | Article | MR 1675206 | Zbl 1053.35124

[25] Lu, K.; Pan, X-B. Surface nucleation of superconductivity in 3-dimension, J. of Differential Equations, Tome 168 (2000) no. 2, pp. 386-452 | Article | MR 1808455 | Zbl 0972.35152

[26] Pino, M. Del; Felmer, P.L.; Sternberg, P. Boundary concentration for eigenvalue problems related to the onset of superconductivity, Comm. Math. Phys., Tome 210 (2000), pp. 413-446 | Article | MR 1776839 | Zbl 0982.35077

[27] Reed, M.; Simon, B. Methods of modern Mathematical Physics, IV: Analysis of operators, Academic Press, New York (1978) | MR 493421 | Zbl 0401.47001

[28] Robert, D. Autour de l’approximation semi-classique, Birkhäuser, Boston (1987) | MR 897108 | Zbl 0621.35001

[29] Saint-James, D.; Sarma, G.; Thomas, E.J. Type II Superconductivity, Pergamon, Oxford (1969)

[30] Simon, B. Semi-classical analysis of low lying eigenvalues I, Ann. Inst. H. Poincaré (Section Physique Théorique), Tome 38 (1983) no. 4, pp. 295-307 | Numdam | MR 708966 | Zbl 0526.35027

[31] Sjöstrand, J. Operators of principal type with interior boundary conditions, Acta Math., Tome 130 (1973), pp. 1-51 | Article | MR 436226 | Zbl 0253.35076

[32] Sternberg, P.; Berger, J.; Rubinstein, J. On the Normal/Superconducting Phase Transition in the Presence of Large Magnetic Fields, Connectivity and Superconductivity, Springer Verlag (Lect. Notes in Physics) Tome M 62 (2000), pp. 188-199 | Zbl 0976.82056

[33] Tilley, D.R.; Tilley, J. Superfluidity and superconductivity, Institute of Physics Publishing, Bristol and Philadelphia (1990)

[34] Tinkham, M. Introduction to Superconductivity, McGraw-Hill Inc, New York (1975)