Cet article s’intéresse au laplacien avec champ magnétique discontinu dans la limite semi-classique. Le champ est supposé prendre exactement deux valeurs non nulles de signes opposés et changer de signe le long d’une courbe fermée et régulière, la « frontière magnétique ». Nous établissons diverses asymptotiques spectrales à l’aide d’une réduction de dimension mettant en jeu une localisation dans l’espace des phases et permettant de traiter la discontinuité du champ magnétique.
The semiclassical Laplacian with discontinuous magnetic field is considered in two dimensions. The magnetic field is sign changing with exactly two distinct values and is discontinuous along a smooth closed curve, thereby producing an attractive magnetic edge. Various accurate spectral asymptotics are established by means of a dimensional reduction involving a microlocal phase space localization allowing to deal with the discontinuity of the field.
CC-BY 4.0
@article{JEP_2023__10__917_0,
author = {S{\o}ren Fournais and Bernard Helffer and Ayman Kachmar and Nicolas Raymond},
title = {Effective operators on an attractive magnetic edge},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {917--944},
publisher = {\'Ecole polytechnique},
volume = {10},
year = {2023},
doi = {10.5802/jep.236},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.236/}
}
TY - JOUR AU - Søren Fournais AU - Bernard Helffer AU - Ayman Kachmar AU - Nicolas Raymond TI - Effective operators on an attractive magnetic edge JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 917 EP - 944 VL - 10 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.236/ DO - 10.5802/jep.236 LA - en ID - JEP_2023__10__917_0 ER -
%0 Journal Article %A Søren Fournais %A Bernard Helffer %A Ayman Kachmar %A Nicolas Raymond %T Effective operators on an attractive magnetic edge %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 917-944 %V 10 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.236/ %R 10.5802/jep.236 %G en %F JEP_2023__10__917_0
Søren Fournais; Bernard Helffer; Ayman Kachmar; Nicolas Raymond. Effective operators on an attractive magnetic edge. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 917-944. doi : 10.5802/jep.236. https://jep.centre-mersenne.org/articles/10.5802/jep.236/
[1] - “Semi-classical eigenvalue estimates under magnetic steps”, 2021, to appear in Anal. PDE | arXiv
[2] - “Lowest energy band function for magnetic steps”, J. Spectral Theory 12 (2022) no. 2, p. 813-833 | DOI | MR | Zbl
[3] - “On the Dirac bag model in strong magnetic fields”, 2020, to appear in Pure and Applied Analysis | HAL
[4] - “On the fundamental state energy for a Schrödinger operator with magnetic field in domains with corners”, Asymptot. Anal. 41 (2005) no. 3-4, p. 215-258 | Zbl
[5] - “Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners”, Ann. Henri Poincaré 7 (2006) no. 5, p. 899-931 | DOI | Zbl
[6] - “Purely magnetic tunneling effect in two dimensions”, Invent. Math. 227 (2022) no. 2, p. 745-793 | DOI | MR | Zbl
[7] - “Edge states for the Robin magnetic Laplacian” (2022), Manuscript in preparation
[8] - “Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian”, Ann. Inst. Fourier (Grenoble) 56 (2006) no. 1, p. 1-67 | DOI | Numdam | MR | Zbl
[9] - Spectral methods in surface superconductivity, Progress in Nonlinear Differential Equations and their Appl., vol. 77, Birkhäuser Boston, Inc., Boston, MA, 2010 | DOI
[10] - “Tunneling effect induced by a curved magnetic edge”, in The physics and mathematics of Elliott Lieb—the 90th anniversary. Vol. I, EMS Press, Berlin, 2022, p. 315-350 | DOI | Zbl
[11] - “On the energy of bound states for magnetic Schrödinger operators”, J. London Math. Soc. (2) 80 (2009) no. 1, p. 233-255 | DOI | Zbl
[12] - “On the asymptotic number of edge states for magnetic Schrödinger operators”, Proc. London Math. Soc. (3) 95 (2007) no. 1, p. 1-19 | DOI | Zbl
[13] - “A mathematical approach to the effective Hamiltonian in perturbed periodic problems”, Comm. Math. Phys. 142 (1991) no. 2, p. 217-244 | DOI | MR | Zbl
[14] - “Edge states for generalized Iwatsuka models: magnetic fields having a fast transition across a curve”, Ann. Henri Poincaré 24 (2023) no. 1, p. 73-105 | DOI | Zbl
[15] - “Magnetic bottles in connection with superconductivity”, J. Funct. Anal. 185 (2001) no. 2, p. 604-680 | DOI | MR | Zbl
[16] - “Multiple wells in the semi-classical limit. I”, Comm. Partial Differential Equations 9 (1984), p. 337-408 | DOI | Zbl
[17] - “Puits multiples en mécanique semi-classique. V. Étude des minipuits”, in Current topics in partial differential equations, Kinokuniya, Tokyo, 1986, p. 133-186 | Zbl
[18] - “Puits multiples en mécanique semi-classique. VI. Cas des puits sous-variétés”, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987) no. 4, p. 353-372 | Numdam | Zbl
[19] - “Band functions in the presence of magnetic steps”, Math. Models Methods Appl. Sci. 26 (2016) no. 1, p. 161-184 | DOI | MR | Zbl
[20] - “Spectral asymptotics for magnetic Schrödinger operators in domains with corners”, J. Spectral Theory 3 (2013) no. 4, p. 553-574 | DOI | Zbl
[21] - Formules de Weyl par réduction de dimension. Applications à des laplaciens électro-magnétiques, Ph. D. Thesis, Université de Rennes 1, 2018
[22] - “A general effective Hamiltonian method”, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18 (2007) no. 3, p. 269-277 | DOI | MR | Zbl
[23] - Bound states of the magnetic Schrödinger operator, EMS Tracts in Math., vol. 27, European Mathematical Society, Zürich, 2017 | DOI
[24] - “The bound state of weakly coupled Schrödinger operators in one and two dimensions”, Ann. Physics 97 (1976) no. 2, p. 279-288 | DOI | Zbl
[25] - Semiclassical analysis, Graduate Studies in Math., vol. 138, American Mathematical Society, Providence, RI, 2012 | DOI
Cité par Sources :