Asymptotic of the smallest eigenvalues of the continuous Anderson Hamiltonian in d≤3
Hsu, Yueh-Sheng; Labbé, Cyril (2022), Asymptotic of the smallest eigenvalues of the continuous Anderson Hamiltonian in d≤3, Stochastics and Partial Differential Equations: Analysis and Computations, p. 26. 10.1007/s40072-022-00252-y
TypeArticle accepté pour publication ou publié
Journal nameStochastics and Partial Differential Equations: Analysis and Computations
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CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Laboratoire de Probabilités, Statistique et Modélisation [LPSM (UMR_8001)]
Abstract (EN)We consider the continuous Anderson Hamiltonian with white noise potential on (−L/2,L/2)d in dimension d≤3, and derive the asymptotic of the smallest eigenvalues when L goes to infinity. We show that these eigenvalues go to −∞ at speed (logL)1/(2−d/2) and identify the prefactor in terms of the optimal constant of the Gagliardo-Nirenberg inequality. This result was already known in dimensions 1 and 2, but appears to be new in dimension 3. We present some conjectures on the fluctuations of the eigenvalues and on the asymptotic shape of the corresponding eigenfunctions near their localisation centers.
Subjects / KeywordsAnderson Hamiltonian; regularity structures; white noise; Schrödinger operator; GagliardoNirenberg inequality
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