On définit l’opérateur de Schrödinger avec champ magnétique sur un graphe infini par la donnée d’un champ magnétique, de poids sur les sommets et de poids sur les arêtes. Lorsque le graphe est de degré borné, on étudie le caractère essentiellement auto-adjoint d’un tel opérateur. Le résultat principal est une version discrète d’un résultat de deux des auteurs du présent article.
We define the magnetic Schrödinger operator on an infinite graph by the data of a magnetic field, some weights on vertices and some weights on edges. We discuss essential self-adjointness of this operator for graphs of bounded degree. The main result is a discrete version of a result of two authors of the present paper.
@article{AFST_2011_6_20_3_599_0,
author = {Colin de Verdi\`ere, Yves and Torki-Hamza, Nabila and Truc, Fran\c{c}oise},
title = {Essential self-adjointness for combinatorial {Schr\"odinger} operators {III-} {Magnetic} fields},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {599--611},
publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 20},
number = {3},
year = {2011},
doi = {10.5802/afst.1319},
zbl = {1250.47025},
mrnumber = {2894840},
language = {en},
url = {http://www.numdam.org/articles/10.5802/afst.1319/}
}
TY - JOUR AU - Colin de Verdière, Yves AU - Torki-Hamza, Nabila AU - Truc, Françoise TI - Essential self-adjointness for combinatorial Schrödinger operators III- Magnetic fields JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2011 SP - 599 EP - 611 VL - 20 IS - 3 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://www.numdam.org/articles/10.5802/afst.1319/ DO - 10.5802/afst.1319 LA - en ID - AFST_2011_6_20_3_599_0 ER -
%0 Journal Article %A Colin de Verdière, Yves %A Torki-Hamza, Nabila %A Truc, Françoise %T Essential self-adjointness for combinatorial Schrödinger operators III- Magnetic fields %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2011 %P 599-611 %V 20 %N 3 %I Université Paul Sabatier, Institut de mathématiques %C Toulouse %U http://www.numdam.org/articles/10.5802/afst.1319/ %R 10.5802/afst.1319 %G en %F AFST_2011_6_20_3_599_0
Colin de Verdière, Yves; Torki-Hamza, Nabila; Truc, Françoise. Essential self-adjointness for combinatorial Schrödinger operators III- Magnetic fields. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. 3, pp. 599-611. doi : 10.5802/afst.1319. http://www.numdam.org/articles/10.5802/afst.1319/
[1] Biggs (N.).— Algebraic Graph Theory, Cambridge University Press (1974). | MR | Zbl
[2] Braverman (M.), Milatovic (O.) & Shubin (M.).— Essential self-adjointness of Schrödinger-type operators on manifolds, Russian Math. Surveys 57, p. 641-692 (2002). | MR | Zbl
[3] Colin de Verdière (Y.).— Spectre de graphes, Cours spécialisés 4, Société mathématique de France (1998). | Zbl
[4] Colin de Verdière (Y.).— Asymptotique de Weyl pour les bouteilles magnétiques, Commun. Math. Phys. 105 p. 327-335 (1986). | MR | Zbl
[5] Colin de Verdière (Y.).— Multiplicities of eigenvalues and tree-width of graphs, J. Combin. Theory Ser. B, 74 p. 121-146 (1998). | MR | Zbl
[6] Colin de Verdière (Y.) & Truc (F.).— Confining quantum particles with a purely magnetic field, Ann. Inst. Fourier (Grenoble), 60 (7) p. 2333-2356 (2010).
[7] Colin de Verdière (Y.), Torki-Hamza (N.) & Truc (F.).— Essential self-adjointness for combinatorial Schrödinger operators II. Metrically non complete graphs, Math. Phys. Anal. Geom. 14 (1) p. 21-38 (2011). | MR
[8] Dodziuk (J.).— Elliptic operators on infinite graphs, Analysis geometry and topology of elliptic operators, 353-368, World Sc. Publ., Hackensack NJ. (2006). | MR | Zbl
[9] Dunford (N.) & Schwartz (J. T.).— Linear operator II, Spectral Theory, John Wiley & Sons, New York (1971). | MR | Zbl
[10] Lieb (E.) & Loss (M.).— Fluxes, Laplacians, and Kasteleyn’s theorem, Duke Math. J., 71 p. 337-363 (1993). | MR | Zbl
[11] Milatovic (O.).— Essential self-adjointness of discrete magnetic Schrödinger operators, ArXiv:1105.3129v1 [math.SP](2011).
[12] Milatovic (O.).— Essential Self-adjointness of magnetic Schrödinger operators on locally finite graphs, Integral Equations and Operator Theory, 71 (1) p.13-27 (2011). | Zbl
[13] Nenciu (G.) & Nenciu (I.).— On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in , Ann. Henri Poincaré, 10 p. 377-394 (2009). | MR | Zbl
[14] Oleinik (I.M.).— On the essential self-adjointness of the Schrödinger operator on complete Riemannian manifolds, Mathematical Notes 54 (3) p. 934-939 (1993). | Zbl
[15] Reed (M.) & Simon (B.).— Methods of Modern mathematical Physics I, Functional analysis, (1980), II, Fourier analysis, Self-adjointness (1975), New York Academic Press. | MR | Zbl
[16] Shubin (M).— Essential self-adjointness for semi-bounded magnetic Schrödinger operators on non-compact manifolds, J. Func. Anal. 186 p. 92-116 (2001). | MR | Zbl
[17] Shubin (M.).— Classical and quantum completness for the Schrödinger operators on non-compact manifolds, Geometric Aspects of Partial Differential Equations (Proc. Sympos., Roskilde, Denmark (1998)) Amer. Math. Soc. Providence, RI, p. 257-269 (1999). | MR | Zbl
[18] Torki-Hamza (N.).— Laplaciens de graphes infinis I- Graphes métriquement complets, Confluentes Mathematici, 2 (3) p. 333-350 (2010). | MR | Zbl
[19] Torki-Hamza (N.).— Essential self-adjointness for combinatorial Schrödinger operators I- Metrically complete graphs, submitted in IWPM 2011, translation in English of [18]. | Zbl
[20] Torki-Hamza (N.).— Stabilité des valeurs propres avec champ magnétique sur une variété Riemannienne et sur un graphe, Thèse de doctorat de l’Université de Grenoble I, France, http://tel.archives-ouvertes.fr/tel-00555758/en/, (1989).
[21] Wojiechowski (R.K.).— Stochastic completeness of graphs, Ph.D. Thesis, The graduate Center of the University of New-York (2008).
Cité par Sources :
