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.
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.
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Keywords: Magnetic Laplacian, discontinuous magnetic field, semiclassical analysis, spectrum
Mot clés : Laplacien magnétique, champ magnétique discontinu, semi-classique, spectre
Søren Fournais 1; Bernard Helffer 2; Ayman Kachmar 3; Nicolas Raymond 4
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, Volume 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
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