Confining quantum particles with a purely magnetic field
Annales de l'Institut Fourier, Volume 60 (2010) no. 7, p. 2333-2356

We consider a Schrödinger operator with a magnetic field (and no electric field) on a domain in the Euclidean space with a compact boundary. We give sufficient conditions on the behaviour of the magnetic field near the boundary which guarantees essential self-adjointness of this operator. From the physical point of view, it means that the quantum particle is confined in the domain by the magnetic field. We construct examples in the case where the boundary is smooth as well as for polytopes; These examples are highly simplified models of what is done for nuclear fusion in tokamacs. We also present some open problems.

Nous considérons un ouvert à bord compact d’un espace euclidien et un opérateur de Schrödinger avec champ magnétique dans cet ouvert. Nous donnons des conditions suffisantes sur la croissance du champ magnétique près du bord qui assurent que l’opérateur de Schrödinger est essentiellement auto-adjoint. Du point de vue de la physique, cela signifie que la particule quantique est confinée dans l’ouvert par le champ magnétique. Nous construisons des exemples dans les polytopes et dans des ouverts à frontières lisses  ; ces exemples de “bouteilles magnétiques” sont des modèles extrêmement simplifiés de ce qui est nécessaire pour la fusion nucléaire dans les tokamacs. Nous présentons aussi des problèmes ouverts.

DOI : https://doi.org/10.5802/aif.2609
Classification:  35J10,  35J25,  35P05,  35Q40,  46N55
Keywords: Magnetic field, Schrödinger operator, self-adjointness
@article{AIF_2010__60_7_2333_0,
     author = {Colin de Verdi\`ere, Yves and Truc, Fran\c coise},
     title = {Confining quantum particles with a purely magnetic field},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {7},
     year = {2010},
     pages = {2333-2356},
     doi = {10.5802/aif.2609},
     mrnumber = {2848672},
     zbl = {1251.81040},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_7_2333_0}
}
Colin de Verdière, Yves; Truc, Françoise. Confining quantum particles with a purely magnetic field. Annales de l'Institut Fourier, Volume 60 (2010) no. 7, pp. 2333-2356. doi : 10.5802/aif.2609. http://www.numdam.org/item/AIF_2010__60_7_2333_0/

[1] Agmon, Shmuel Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N -body Schrödinger operators, Princeton University Press, Princeton, NJ, Mathematical Notes, Tome 29 (1982) | MR 745286 | Zbl 0503.35001

[2] Alexandroff, Paul; Hopf, Heinz Topologie. Band I, Chelsea Publishing Co., Bronx, N. Y. (1972) | MR 396210

[3] Avron, J.; Herbst, I.; Simon, B. Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J., Tome 45 (1978) no. 4, pp. 847-883 | Article | MR 518109 | Zbl 0399.35029

[4] Balinsky, Alexander; Laptev, Ari; Sobolev, Alexander V. Generalized Hardy inequality for the magnetic Dirichlet forms, J. Statist. Phys., Tome 116 (2004) no. 1-4, pp. 507-521 | Article | MR 2083152 | Zbl 1127.26015

[5] Colin De Verdière, Yves L’asymptotique de Weyl pour les bouteilles magnétiques, Comm. Math. Phys., Tome 105 (1986) no. 2, pp. 327-335 | Article | MR 849211 | Zbl 0612.35102

[6] Cycon, H. L.; Froese, R. G.; Kirsch, W.; Simon, B. Schrödinger operators with application to quantum mechanics and global geometry, Springer-Verlag, Berlin, Texts and Monographs in Physics (1987) | MR 883643 | Zbl 0619.47005

[7] Dufresnoy, Alain Un exemple de champ magnétique dans R ν , Duke Math. J., Tome 50 (1983) no. 3, pp. 729-734 | Article | MR 714827 | Zbl 0532.35021

[8] Dunford, Nelson; Schwartz, Jacob T. Linear operators. Part III: Spectral operators, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney (1971) (With the assistance of William G. Bade and Robert G. Bartle, Pure and Applied Mathematics, Vol. VII) | MR 412888 | Zbl 0635.47003

[9] Erdős, László; Solovej, Jan Philip Semiclassical eigenvalue estimates for the Pauli operator with strong nonhomogeneous magnetic fields. I. Nonasymptotic Lieb-Thirring-type estimate, Duke Math. J., Tome 96 (1999) no. 1, pp. 127-173 | Article | MR 1663923 | Zbl 1047.81022

[10] Erdős, László; Solovej, Jan Philip Magnetic Lieb-Thirring inequalities with optimal dependence on the field strength, J. Statist. Phys., Tome 116 (2004) no. 1-4, pp. 475-506 | Article | MR 2083151 | Zbl 1138.81017

[11] Erdős, László; Solovej, Jan Philip Uniform Lieb-Thirring inequality for the three-dimensional Pauli operator with a strong non-homogeneous magnetic field, Ann. Henri Poincaré, Tome 5 (2004) no. 4, pp. 671-741 | Article | MR 2090449 | Zbl 1054.81016

[12] Guillemin, Victor; Pollack, Alan Differential topology, Prentice-Hall Inc., Englewood Cliffs, N.J. (1974) | MR 348781 | Zbl 0361.57001

[13] Heinonen, Juha Lectures on Lipschitz analysis, University of Jyväskylä, Jyväskylä, Report. University of Jyväskylä Department of Mathematics and Statistics, Tome 100 (2005) (http://www.math.jyu.fi/tutkimus/ber.html) | MR 2177410 | Zbl 1086.30003

[14] http://en.wikipedia.org/wiki/Tokamak

[15] Ikebe, Teruo; Kato, Tosio Uniqueness of the self-adjoint extension of singular elliptic differential operators, Arch. Rational Mech. Anal., Tome 9 (1962), pp. 77-92 | Article | MR 142894 | Zbl 0103.31801

[16] Kalf, H.; Schmincke, U.-W.; Walter, J.; Wüst, R. On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Springer, Berlin (1975), p. 182-226. Lecture Notes in Math., Vol. 448 | MR 397192 | Zbl 0311.47021

[17] Kostant, Bertram Quantization and unitary representations. I. Prequantization, Lectures in modern analysis and applications, III, Springer, Berlin (1970), p. 87-208. Lecture Notes in Math., Vol. 170 | MR 294568 | Zbl 0223.53028

[18] Kuwabara, Ruishi On spectra of the Laplacian on vector bundles, J. Math. Tokushima Univ., Tome 16 (1982), pp. 1-23 | MR 691445 | Zbl 0504.53039

[19] Kuwabara, Ruishi Spectrum of the Schrödinger operator on a line bundle over complex projective spaces, Tohoku Math. J. (2), Tome 40 (1988) no. 2, pp. 199-211 | Article | MR 943819 | Zbl 0652.53044

[20] Morrey, Charles B. Jr. Multiple integrals in the calculus of variations, Springer-Verlag New York, Inc., New York, Die Grundlehren der mathematischen Wissenschaften, Band 130 (1966) | MR 202511 | Zbl 0142.38701

[21] Nenciu, Gheorghe; Nenciu, Irina On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in n , Ann. Henri Poincaré, Tome 10 (2009) no. 2, pp. 377-394 | Article | MR 2511891 | Zbl 1205.81088

[22] Nenciu, Gheorghe; Nenciu, Irina Remarks on essential self-adjointness for magnetic Schrödinger and Pauli operators on bounded domains in 2 (2010) (arXiv:1003.3099)

[23] Rademacher, Hans Über partielle und totale differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale, Math. Ann., Tome 79 (1919) no. 4, pp. 340-359 | Article | MR 1511935

[24] Reed, Michael; Simon, Barry Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich Publishers], New York (1975) | MR 493420 | Zbl 0242.46001

[25] Shubin, Mikhail Essential self-adjointness for semi-bounded magnetic Schrödinger operators on non-compact manifolds, J. Funct. Anal., Tome 186 (2001) no. 1, pp. 92-116 | Article | MR 1863293 | Zbl 0997.58021

[26] Sigal, I. M. Geometric methods in the quantum many-body problem. Nonexistence of very negative ions, Comm. Math. Phys., Tome 85 (1982) no. 2, pp. 309-324 | Article | MR 676004 | Zbl 0503.47041

[27] Simon, Barry Essential self-adjointness of Schrödinger operators with singular potentials, Arch. Rational Mech. Anal., Tome 52 (1973), pp. 44-48 | Article | MR 338548 | Zbl 0277.47007

[28] Simon, Barry Schrödinger operators with singular magnetic vector potentials, Math. Z., Tome 131 (1973), pp. 361-370 | Article | MR 322336 | Zbl 0277.47006

[29] Torki-Hamza, Nabila Stabilité des valeurs propres et champ magnétique sur une variété riemannienne et sur un graphe, Grenoble University (1989) (Ph. D. Thesis)

[30] Truc, Françoise Trajectoires bornées d’une particule soumise à un champ magnétique symétrique linéaire, Ann. Inst. H. Poincaré Phys. Théor., Tome 64 (1996) no. 2, pp. 127-154 | Numdam | MR 1386214 | Zbl 0862.70005

[31] Truc, Françoise Semi-classical asymptotics for magnetic bottles, Asymptot. Anal., Tome 15 (1997) no. 3-4, pp. 385-395 | MR 1487718 | Zbl 0902.35079