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Yves Colin de Verdière; Françoise Truc
Confining quantum particles with a purely magnetic field
(Confinement de particules quantiques avec un champ magnétique)
Annales de l'institut Fourier, 60 no. 7 (2010), p. 2333-2356, doi: 10.5802/aif.2609
Article PDF | Reviews MR 2848672 | Zbl 1251.81040
Class. Math.: 35J10, 35J25, 35P05, 35Q40, 46N55
Keywords: Magnetic field, Schrödinger operator, self-adjointness
[1] Shmuel Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of $N$-body Schrödinger operators, Mathematical Notes 29, Princeton University Press, Princeton, NJ, 1982 MR 745286 | Zbl 0503.35001
[2] Paul Alexandroff & Heinz Hopf, Topologie. Band I, Chelsea Publishing Co., Bronx, N. Y., 1972 MR 396210
[3] J. Avron, I. Herbst & B. Simon, “Schrödinger operators with magnetic fields. I. General interactions”, Duke Math. J. 45 (1978) no. 4, p. 847-883 Article | MR 518109 | Zbl 0399.35029
[4] Alexander Balinsky, Ari Laptev & Alexander V. Sobolev, “Generalized Hardy inequality for the magnetic Dirichlet forms”, J. Statist. Phys. 116 (2004) no. 1-4, p. 507-521 Article | MR 2083152 | Zbl 1127.26015
[5] Yves Colin de Verdière, “L’asymptotique de Weyl pour les bouteilles magnétiques”, Comm. Math. Phys. 105 (1986) no. 2, p. 327-335 Article | MR 849211 | Zbl 0612.35102
[6] H. L. Cycon, R. G. Froese, W. Kirsch & B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987 MR 883643 | Zbl 0619.47005
[7] Alain Dufresnoy, “Un exemple de champ magnétique dans ${\bf R}^{\nu }$”, Duke Math. J. 50 (1983) no. 3, p. 729-734 Article | MR 714827 | Zbl 0532.35021
[8] Nelson Dunford & Jacob T. Schwartz, 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] László Erdős & Jan Philip Solovej, “Semiclassical eigenvalue estimates for the Pauli operator with strong nonhomogeneous magnetic fields. I. Nonasymptotic Lieb-Thirring-type estimate”, Duke Math. J. 96 (1999) no. 1, p. 127-173 Article | MR 1663923 | Zbl 1047.81022
[10] László Erdős & Jan Philip Solovej, “Magnetic Lieb-Thirring inequalities with optimal dependence on the field strength”, J. Statist. Phys. 116 (2004) no. 1-4, p. 475-506 Article | MR 2083151 | Zbl 1138.81017
[11] László Erdős & Jan Philip Solovej, “Uniform Lieb-Thirring inequality for the three-dimensional Pauli operator with a strong non-homogeneous magnetic field”, Ann. Henri Poincaré 5 (2004) no. 4, p. 671-741 Article | MR 2090449 | Zbl 1054.81016
[12] Victor Guillemin & Alan Pollack, Differential topology, Prentice-Hall Inc., Englewood Cliffs, N.J., 1974 MR 348781 | Zbl 0361.57001
[13] Juha Heinonen, Lectures on Lipschitz analysis, Report. University of Jyväskylä Department of Mathematics and Statistics 100, University of Jyväskylä, Jyväskylä, 2005, http://www.math.jyu.fi/tutkimus/ber.html MR 2177410 | Zbl 1086.30003
[14] , http://en.wikipedia.org/wiki/Tokamak
[15] Teruo Ikebe & Tosio Kato, “Uniqueness of the self-adjoint extension of singular elliptic differential operators”, Arch. Rational Mech. Anal. 9 (1962), p. 77-92 Article | MR 142894 | Zbl 0103.31801
[16] H. Kalf, U.-W. Schmincke, J. Walter & R. Wüst, 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, 1975, p. 182–226. Lecture Notes in Math., Vol. 448 MR 397192 | Zbl 0311.47021
[17] Bertram Kostant, Quantization and unitary representations. I. Prequantization, Lectures in modern analysis and applications, III, Springer, 1970, p. 87–208. Lecture Notes in Math., Vol. 170 MR 294568 | Zbl 0223.53028
[18] Ruishi Kuwabara, “On spectra of the Laplacian on vector bundles”, J. Math. Tokushima Univ. 16 (1982), p. 1-23 MR 691445 | Zbl 0504.53039
[19] Ruishi Kuwabara, “Spectrum of the Schrödinger operator on a line bundle over complex projective spaces”, Tohoku Math. J. (2) 40 (1988) no. 2, p. 199-211 Article | MR 943819 | Zbl 0652.53044
[20] Charles B. Morrey, Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966 MR 202511 | Zbl 0142.38701
[21] Gheorghe Nenciu & Irina Nenciu, “On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in $\mathbb{ R}^n$”, Ann. Henri Poincaré 10 (2009) no. 2, p. 377-394 Article | MR 2511891 | Zbl 1205.81088
[22] Gheorghe Nenciu & Irina Nenciu, “Remarks on essential self-adjointness for magnetic Schrödinger and Pauli operators on bounded domains in $\mathbb{ R}^2$”, arXiv:1003.3099, 2010
[23] Hans Rademacher, “Über partielle und totale differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale”, Math. Ann. 79 (1919) no. 4, p. 340-359 Article | MR 1511935 | JFM 47.0243.01
[24] Michael Reed & Barry Simon, 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] Mikhail Shubin, “Essential self-adjointness for semi-bounded magnetic Schrödinger operators on non-compact manifolds”, J. Funct. Anal. 186 (2001) no. 1, p. 92-116 Article | MR 1863293 | Zbl 0997.58021
[26] I. M. Sigal, “Geometric methods in the quantum many-body problem. Nonexistence of very negative ions”, Comm. Math. Phys. 85 (1982) no. 2, p. 309-324 Article | MR 676004 | Zbl 0503.47041
[27] Barry Simon, “Essential self-adjointness of Schrödinger operators with singular potentials”, Arch. Rational Mech. Anal. 52 (1973), p. 44-48 Article | MR 338548 | Zbl 0277.47007
[28] Barry Simon, “Schrödinger operators with singular magnetic vector potentials”, Math. Z. 131 (1973), p. 361-370 Article | MR 322336 | Zbl 0277.47006
[29] Nabila Torki-Hamza, Stabilité des valeurs propres et champ magnétique sur une variété riemannienne et sur un graphe, Ph. D. Thesis, Grenoble University, 1989
[30] Françoise Truc, “Trajectoires bornées d’une particule soumise à un champ magnétique symétrique linéaire”, Ann. Inst. H. Poincaré Phys. Théor. 64 (1996) no. 2, p. 127-154 Numdam | MR 1386214 | Zbl 0862.70005
[31] Françoise Truc, “Semi-classical asymptotics for magnetic bottles”, Asymptot. Anal. 15 (1997) no. 3-4, p. 385-395 MR 1487718 | Zbl 0902.35079
Yves Colin de Verdière; Françoise Truc
Confining quantum particles with a purely magnetic field
(Confinement de particules quantiques avec un champ magnétique)
Annales de l'institut Fourier, 60 no. 7 (2010), p. 2333-2356, doi: 10.5802/aif.2609
Article PDF | Reviews MR 2848672 | Zbl 1251.81040
Class. Math.: 35J10, 35J25, 35P05, 35Q40, 46N55
Keywords: Magnetic field, Schrödinger operator, self-adjointness
Résumé - Abstract
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.
Bibliography
[2] Paul Alexandroff & Heinz Hopf, Topologie. Band I, Chelsea Publishing Co., Bronx, N. Y., 1972 MR 396210
[3] J. Avron, I. Herbst & B. Simon, “Schrödinger operators with magnetic fields. I. General interactions”, Duke Math. J. 45 (1978) no. 4, p. 847-883 Article | MR 518109 | Zbl 0399.35029
[4] Alexander Balinsky, Ari Laptev & Alexander V. Sobolev, “Generalized Hardy inequality for the magnetic Dirichlet forms”, J. Statist. Phys. 116 (2004) no. 1-4, p. 507-521 Article | MR 2083152 | Zbl 1127.26015
[5] Yves Colin de Verdière, “L’asymptotique de Weyl pour les bouteilles magnétiques”, Comm. Math. Phys. 105 (1986) no. 2, p. 327-335 Article | MR 849211 | Zbl 0612.35102
[6] H. L. Cycon, R. G. Froese, W. Kirsch & B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987 MR 883643 | Zbl 0619.47005
[7] Alain Dufresnoy, “Un exemple de champ magnétique dans ${\bf R}^{\nu }$”, Duke Math. J. 50 (1983) no. 3, p. 729-734 Article | MR 714827 | Zbl 0532.35021
[8] Nelson Dunford & Jacob T. Schwartz, 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] László Erdős & Jan Philip Solovej, “Semiclassical eigenvalue estimates for the Pauli operator with strong nonhomogeneous magnetic fields. I. Nonasymptotic Lieb-Thirring-type estimate”, Duke Math. J. 96 (1999) no. 1, p. 127-173 Article | MR 1663923 | Zbl 1047.81022
[10] László Erdős & Jan Philip Solovej, “Magnetic Lieb-Thirring inequalities with optimal dependence on the field strength”, J. Statist. Phys. 116 (2004) no. 1-4, p. 475-506 Article | MR 2083151 | Zbl 1138.81017
[11] László Erdős & Jan Philip Solovej, “Uniform Lieb-Thirring inequality for the three-dimensional Pauli operator with a strong non-homogeneous magnetic field”, Ann. Henri Poincaré 5 (2004) no. 4, p. 671-741 Article | MR 2090449 | Zbl 1054.81016
[12] Victor Guillemin & Alan Pollack, Differential topology, Prentice-Hall Inc., Englewood Cliffs, N.J., 1974 MR 348781 | Zbl 0361.57001
[13] Juha Heinonen, Lectures on Lipschitz analysis, Report. University of Jyväskylä Department of Mathematics and Statistics 100, University of Jyväskylä, Jyväskylä, 2005, http://www.math.jyu.fi/tutkimus/ber.html MR 2177410 | Zbl 1086.30003
[14] , http://en.wikipedia.org/wiki/Tokamak
[15] Teruo Ikebe & Tosio Kato, “Uniqueness of the self-adjoint extension of singular elliptic differential operators”, Arch. Rational Mech. Anal. 9 (1962), p. 77-92 Article | MR 142894 | Zbl 0103.31801
[16] H. Kalf, U.-W. Schmincke, J. Walter & R. Wüst, 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, 1975, p. 182–226. Lecture Notes in Math., Vol. 448 MR 397192 | Zbl 0311.47021
[17] Bertram Kostant, Quantization and unitary representations. I. Prequantization, Lectures in modern analysis and applications, III, Springer, 1970, p. 87–208. Lecture Notes in Math., Vol. 170 MR 294568 | Zbl 0223.53028
[18] Ruishi Kuwabara, “On spectra of the Laplacian on vector bundles”, J. Math. Tokushima Univ. 16 (1982), p. 1-23 MR 691445 | Zbl 0504.53039
[19] Ruishi Kuwabara, “Spectrum of the Schrödinger operator on a line bundle over complex projective spaces”, Tohoku Math. J. (2) 40 (1988) no. 2, p. 199-211 Article | MR 943819 | Zbl 0652.53044
[20] Charles B. Morrey, Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966 MR 202511 | Zbl 0142.38701
[21] Gheorghe Nenciu & Irina Nenciu, “On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in $\mathbb{ R}^n$”, Ann. Henri Poincaré 10 (2009) no. 2, p. 377-394 Article | MR 2511891 | Zbl 1205.81088
[22] Gheorghe Nenciu & Irina Nenciu, “Remarks on essential self-adjointness for magnetic Schrödinger and Pauli operators on bounded domains in $\mathbb{ R}^2$”, arXiv:1003.3099, 2010
[23] Hans Rademacher, “Über partielle und totale differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale”, Math. Ann. 79 (1919) no. 4, p. 340-359 Article | MR 1511935 | JFM 47.0243.01
[24] Michael Reed & Barry Simon, 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] Mikhail Shubin, “Essential self-adjointness for semi-bounded magnetic Schrödinger operators on non-compact manifolds”, J. Funct. Anal. 186 (2001) no. 1, p. 92-116 Article | MR 1863293 | Zbl 0997.58021
[26] I. M. Sigal, “Geometric methods in the quantum many-body problem. Nonexistence of very negative ions”, Comm. Math. Phys. 85 (1982) no. 2, p. 309-324 Article | MR 676004 | Zbl 0503.47041
[27] Barry Simon, “Essential self-adjointness of Schrödinger operators with singular potentials”, Arch. Rational Mech. Anal. 52 (1973), p. 44-48 Article | MR 338548 | Zbl 0277.47007
[28] Barry Simon, “Schrödinger operators with singular magnetic vector potentials”, Math. Z. 131 (1973), p. 361-370 Article | MR 322336 | Zbl 0277.47006
[29] Nabila Torki-Hamza, Stabilité des valeurs propres et champ magnétique sur une variété riemannienne et sur un graphe, Ph. D. Thesis, Grenoble University, 1989
[30] Françoise Truc, “Trajectoires bornées d’une particule soumise à un champ magnétique symétrique linéaire”, Ann. Inst. H. Poincaré Phys. Théor. 64 (1996) no. 2, p. 127-154 Numdam | MR 1386214 | Zbl 0862.70005
[31] Françoise Truc, “Semi-classical asymptotics for magnetic bottles”, Asymptot. Anal. 15 (1997) no. 3-4, p. 385-395 MR 1487718 | Zbl 0902.35079
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