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.
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.
@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}, volume = {60}, year = {2010}, pages = {2333-2356}, doi = {10.5802/aif.2609}, zbl = {1251.81040}, mrnumber = {2848672}, language = {en}, url = {http://dml.mathdoc.fr/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, Tome 60 (2010) pp. 2333-2356. doi : 10.5802/aif.2609. http://gdmltest.u-ga.fr/item/AIF_2010__60_7_2333_0/
[1] Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of -body Schrödinger operators, Princeton University Press, Princeton, NJ, Mathematical Notes, Tome 29 (1982) | MR 745286 | Zbl 0503.35001
[2] Topologie. Band I, Chelsea Publishing Co., Bronx, N. Y. (1972) | MR 396210
[3] 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] 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] 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] 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] Un exemple de champ magnétique dans , Duke Math. J., Tome 50 (1983) no. 3, pp. 729-734 | Article | MR 714827 | Zbl 0532.35021
[8] 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] 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] 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] 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] Differential topology, Prentice-Hall Inc., Englewood Cliffs, N.J. (1974) | MR 348781 | Zbl 0361.57001
[13] 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] 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] 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] 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] On spectra of the Laplacian on vector bundles, J. Math. Tokushima Univ., Tome 16 (1982), pp. 1-23 | MR 691445 | Zbl 0504.53039
[19] 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] 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] On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in , Ann. Henri Poincaré, Tome 10 (2009) no. 2, pp. 377-394 | Article | MR 2511891 | Zbl 1205.81088
[22] Remarks on essential self-adjointness for magnetic Schrödinger and Pauli operators on bounded domains in (2010) (arXiv:1003.3099)
[23] 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] 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] 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] 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] 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] Schrödinger operators with singular magnetic vector potentials, Math. Z., Tome 131 (1973), pp. 361-370 | Article | MR 322336 | Zbl 0277.47006
[29] Stabilité des valeurs propres et champ magnétique sur une variété riemannienne et sur un graphe, Grenoble University (1989) (Ph. D. Thesis)
[30] 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] Semi-classical asymptotics for magnetic bottles, Asymptot. Anal., Tome 15 (1997) no. 3-4, pp. 385-395 | MR 1487718 | Zbl 0902.35079