[Une construction simple de l’opérateur d’Anderson par sa forme quadratique en dimensions deux et trois]
Nous fournissons une construction simple de l’opérateur d’Anderson en dimensions deux et trois. Cela est réalisé à travers sa forme quadratique. Nous nous appuyons sur une transformation exponentielle au lieu des structures de régularité ou du calcul paracontrôlé, qui sont généralement utilisés pour la construction de l’opérateur. La connaissance de la forme est suffisamment robuste pour déduire des propriétés importantes telles que la positivité et l’irréductibilité du semi-groupe correspondant. Cette dernière propriété permet de démontrer l’existence d’un trou spectral.
We provide a simple construction of the Anderson operator in dimensions two and three. This is done through its quadratic form. We rely on an exponential transform instead of the regularity structures or paracontrolled calculus which are usually used for the construction of the operator. The knowledge of the form is robust enough to deduce important properties such as positivity and irreducibility of the corresponding semigroup. The latter property gives existence of a spectral gap.
Révisé le :
Accepté le :
Publié le :
Keywords: Anderson form, singular stochastic operator, Schrödinger operator, renormalization, positivity, spectral gap
Mots-clés : Forme d’Anderson, opérateurs stochastiques singuliers, opérateur de Schrödinger, renormalisation, positivité, trou spectral
Antoine Mouzard 1 ; El Maati Ouhabaz 2
CC-BY 4.0
@article{CRMATH_2025__363_G2_183_0,
author = {Antoine Mouzard and El Maati Ouhabaz},
title = {A simple construction of the {Anderson} operator via its quadratic form in dimensions two and three},
journal = {Comptes Rendus. Math\'ematique},
pages = {183--197},
publisher = {Acad\'emie des sciences, Paris},
volume = {363},
year = {2025},
doi = {10.5802/crmath.712},
language = {en},
}
TY - JOUR AU - Antoine Mouzard AU - El Maati Ouhabaz TI - A simple construction of the Anderson operator via its quadratic form in dimensions two and three JO - Comptes Rendus. Mathématique PY - 2025 SP - 183 EP - 197 VL - 363 PB - Académie des sciences, Paris DO - 10.5802/crmath.712 LA - en ID - CRMATH_2025__363_G2_183_0 ER -
%0 Journal Article %A Antoine Mouzard %A El Maati Ouhabaz %T A simple construction of the Anderson operator via its quadratic form in dimensions two and three %J Comptes Rendus. Mathématique %D 2025 %P 183-197 %V 363 %I Académie des sciences, Paris %R 10.5802/crmath.712 %G en %F CRMATH_2025__363_G2_183_0
Antoine Mouzard; El Maati Ouhabaz. A simple construction of the Anderson operator via its quadratic form in dimensions two and three. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 183-197. doi : 10.5802/crmath.712. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.712/
[1] The continuous Anderson hamiltonian in dimension two (2015) | arXiv
[2] Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften, 343, Springer, 2011, xvi+523 pages | DOI | MR
[3]
[4] Analysis of the Anderson operator (2023) | arXiv
[5] Renormalising SPDEs in regularity structures, J. Eur. Math. Soc., Volume 23 (2021) no. 3, pp. 869-947 | DOI | MR | Zbl
[6] Algebraic renormalisation of regularity structures, Invent. Math., Volume 215 (2019) no. 3, pp. 1039-1156 | DOI | MR
[7] An analytic BPHZ theorem for regularity structures (2018) | arXiv
[8] The logarithmic Schrödinger equation with spatial white noise on the full space (2023) | arXiv
[9] Asymptotics of the eigenvalues of the Anderson Hamiltonian with white noise potential in two dimensions, Ann. Probab., Volume 49 (2021) no. 4, pp. 1917-1964 | DOI | MR | Zbl
[10] The Schrödinger equation with spatial white noise potential, Electron. J. Probab., Volume 23 (2018), 28, 16 pages | DOI | MR | Zbl
[11] Localization crossover for the continuous Anderson Hamiltonian in 1-d, Invent. Math., Volume 235 (2024) no. 2, pp. 345-440 | DOI | MR | Zbl
[12] On spectra of the Schrödinger operator with a white Gaussian noise potential, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 37 (1977) no. 3, pp. 267-274 | DOI | MR | Zbl
[13] Paracontrolled distributions and singular PDEs, Forum Math. Pi, Volume 3 (2015), e6, 75 pages | DOI | MR | Zbl
[14] KPZ reloaded, Commun. Math. Phys., Volume 349 (2017) no. 1, pp. 165-269 | DOI | MR | Zbl
[15] Semilinear evolution equations for the Anderson Hamiltonian in two and three dimensions, Stoch. Partial Differ. Equ., Anal. Comput., Volume 8 (2020) no. 1, pp. 82-149 | DOI | MR | Zbl
[16] A theory of regularity structures, Invent. Math., Volume 198 (2014) no. 2, pp. 269-504 | DOI | MR | Zbl
[17] A simple construction of the continuum parabolic Anderson model on
[18] A simple construction of the dynamical
[19] The continuous Anderson Hamiltonian in
[20] Anderson Hamiltonians with singular potentials (2024) | arXiv
[21] 2D random magnetic Laplacian with white noise magnetic field, Stochastic Processes Appl., Volume 143 (2022), pp. 160-184 | DOI | MR | Zbl
[22] The infinitesimal generator of the Brox diffusion (2022) | arXiv
[23] Weyl law for the Anderson Hamiltonian on a two-dimensional manifold, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 58 (2022) no. 3, pp. 1385-1425 | DOI | MR | Zbl
[24] Strichartz inequalities with white noise potential on compact surfaces, Anal. PDE, Volume 17 (2024) no. 2, pp. 421-454 | DOI | MR | Zbl
[25] Analysis of heat equations on domains, London Mathematical Society Monographs, 31, Princeton University Press, 2005, xiv+284 pages | MR
[26] On some series of functions. I, II., Proc. Camb. Philos. Soc., Volume 26 (1930), p. 337-357, 458–474 | Zbl
[27] On some series of functions. III, Proc. Camb. Philos. Soc., Volume 28 (1932), pp. 190-205 | Zbl
[28] Two dimensional nonlinear Schrödinger equation with spatial white noise potential and fourth order nonlinearity, Stoch. Partial Differ. Equ., Anal. Comput., Volume 11 (2023) no. 3, pp. 948-987 | DOI | MR | Zbl
Cité par Sources :
Commentaires - Politique