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
[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).
- Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields, Advances in Mathematics, Volume 445 (2024), p. 54 (Id/No 109665) | DOI:10.1016/j.aim.2024.109665 | Zbl:1545.35041
- Pointwise eigenvector estimates by landscape functions: some variations on the Filoche-Mayboroda-van den Berg bound, Mathematische Nachrichten, Volume 297 (2024) no. 5, pp. 1749-1771 | DOI:10.1002/mana.202300239 | Zbl:1541.35162
- Self-adjointness of magnetic Laplacians on triangulations, Filomat, Volume 37 (2023) no. 11, p. 3527 | DOI:10.2298/fil2311527a
- Trace formulas for magnetic Schrödinger operators on periodic graphs and their applications, Linear Algebra and its Applications, Volume 676 (2023), pp. 395-440 | DOI:10.1016/j.laa.2023.07.025 | Zbl:7734093
- , 2022 Days on Diffraction (DD) (2022), p. 1 | DOI:10.1109/dd55230.2022.9961025
- A Glazman-Povzner-Wienholtz theorem on graphs, Advances in Mathematics, Volume 395 (2022), p. 30 (Id/No 108158) | DOI:10.1016/j.aim.2021.108158 | Zbl:1503.81035
- A local test for global extrema in the dispersion relation of a periodic graph, Pure and Applied Analysis, Volume 4 (2022) no. 2, pp. 257-286 | DOI:10.2140/paa.2022.4.257 | Zbl:1505.35312
- The magnetic discrete Laplacian inferred from the Gauß-Bonnet operator and application, Annals of Functional Analysis, Volume 12 (2021) no. 2, p. 30 (Id/No 33) | DOI:10.1007/s43034-021-00119-8 | Zbl:1460.05108
- Self-adjointness of perturbed bi-Laplacians on infinite graphs, Indagationes Mathematicae. New Series, Volume 32 (2021) no. 2, pp. 442-455 | DOI:10.1016/j.indag.2020.12.003 | Zbl:1459.05221
- Uniqueness of form extensions and domination of semigroups, Journal of Functional Analysis, Volume 280 (2021) no. 6, p. 28 (Id/No 108848) | DOI:10.1016/j.jfa.2020.108848 | Zbl:7298632
- Riesz decompositions for Schrödinger operators on graphs, Journal of Mathematical Analysis and Applications, Volume 495 (2021) no. 1, p. 22 (Id/No 124674) | DOI:10.1016/j.jmaa.2020.124674 | Zbl:1469.31023
- Magnetic-sparseness and Schrödinger operators on graphs, Annales Henri Poincaré, Volume 21 (2020) no. 5, pp. 1489-1516 | DOI:10.1007/s00023-020-00885-6 | Zbl:1444.81018
- Discrete magnetic bottles on quasi-linear graphs, Complex Analysis and Operator Theory, Volume 13 (2019) no. 3, pp. 1401-1417 | DOI:10.1007/s11785-018-00883-x | Zbl:1479.05210
- Boundary representation of Dirichlet forms on discrete spaces, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 126 (2019), pp. 109-143 | DOI:10.1016/j.matpur.2018.10.005 | Zbl:1418.31013
- The adjacency matrix and the discrete Laplacian acting on forms, Mathematical Physics, Analysis and Geometry, Volume 22 (2019) no. 1, p. 27 (Id/No 9) | DOI:10.1007/s11040-019-9301-0 | Zbl:1411.81095
- The magnetic Laplacian acting on discrete cusps, Documenta Mathematica, Volume 22 (2017), pp. 1709-1727 | DOI:10.25537/dm.2017v22.1709-1727 | Zbl:1441.05136
- Magnetic Schrödinger operators on periodic discrete graphs, Journal of Functional Analysis, Volume 272 (2017) no. 4, pp. 1625-1660 | DOI:10.1016/j.jfa.2016.12.015 | Zbl:1355.81081
- Spectral and scattering theory for Gauss-Bonnet operators on perturbed topological crystals, Journal of Mathematical Analysis and Applications, Volume 452 (2017) no. 2, pp. 792-813 | DOI:10.1016/j.jmaa.2017.03.002 | Zbl:1365.81054
- Continuity of the spectra for families of magnetic operators on
, Analysis and Mathematical Physics, Volume 6 (2016) no. 4, pp. 327-343 | DOI:10.1007/s13324-015-0121-5 | Zbl:1353.81054 - A Feynman-Kac-Itô formula for magnetic Schrödinger operators on graphs, Probability Theory and Related Fields, Volume 165 (2016) no. 1-2, pp. 365-399 | DOI:10.1007/s00440-015-0633-9 | Zbl:1341.81027
- Essential spectrum and Weyl asymptotics for discrete Laplacians, Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI, Volume 24 (2015) no. 3, pp. 563-624 | DOI:10.5802/afst.1456 | Zbl:1342.81142
- Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians, Calculus of Variations and Partial Differential Equations, Volume 54 (2015) no. 4, pp. 4165-4196 | DOI:10.1007/s00526-015-0935-x | Zbl:1330.05103
- Maximal accretive extensions of Schrödinger operators on vector bundles over infinite graphs, Integral Equations and Operator Theory, Volume 81 (2015) no. 1, pp. 35-52 | DOI:10.1007/s00020-014-2196-z | Zbl:1306.05169
- Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs, Journal of Mathematical Physics, Volume 56 (2015) no. 12, p. 122111 | DOI:10.1063/1.4937119 | Zbl:1335.81078
- Magnetic energies and Feynman-Kac-Itô formulas for symmetric Markov processes, Stochastic Analysis and Applications, Volume 33 (2015) no. 6, pp. 1020-1049 | DOI:10.1080/07362994.2015.1077715 | Zbl:1329.60266
- Self-adjoint extensions of discrete magnetic Schrödinger operators, Annales Henri Poincaré, Volume 15 (2014) no. 5, pp. 917-936 | DOI:10.1007/s00023-013-0261-9 | Zbl:1288.81039
- Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians, Journal of Functional Analysis, Volume 266 (2014) no. 5, pp. 2662-2688 | DOI:10.1016/j.jfa.2013.10.012 | Zbl:1292.35300
- Nodal count of graph eigenfunctions via magnetic perturbation, Analysis PDE, Volume 6 (2013) no. 5, p. 1213 | DOI:10.2140/apde.2013.6.1213
- A spectral property of discrete Schrödinger operators with non-negative potentials, Integral Equations and Operator Theory, Volume 76 (2013) no. 2, pp. 285-300 | DOI:10.1007/s00020-013-2060-6 | Zbl:1273.35104
- Dirac and magnetic Schrödinger operators on fractals, Journal of Functional Analysis, Volume 265 (2013) no. 11, pp. 2830-2854 | DOI:10.1016/j.jfa.2013.07.021 | Zbl:1319.47037
- A note on self-adjoint extensions of the Laplacian on weighted graphs, Journal of Functional Analysis, Volume 265 (2013) no. 8, pp. 1556-1578 | DOI:10.1016/j.jfa.2013.06.004 | Zbl:1435.35400
- A Sears-type self-adjointness result for discrete magnetic Schrödinger operators, Journal of Mathematical Analysis and Applications, Volume 396 (2012) no. 2, pp. 801-809 | DOI:10.1016/j.jmaa.2012.07.028 | Zbl:1268.47057
- Essential self-adjointness of magnetic Schrödinger operators on locally finite graphs, Integral Equations and Operator Theory, Volume 71 (2011) no. 1, pp. 13-27 | DOI:10.1007/s00020-011-1882-3 | Zbl:1234.35077
Cité par 33 documents. Sources : Crossref, zbMATH