The continuous Anderson hamiltonian in d≤3
Labbé, Cyril (2019), The continuous Anderson hamiltonian in d≤3, Journal of Functional Analysis, 277, 9, p. 3187-3235. 10.1016/j.jfa.2019.05.027
TypeArticle accepté pour publication ou publié
Journal nameJournal of Functional Analysis
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CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)We construct the continuous Anderson hamiltonian on (−L,L)d driven by a white noise and endowed with either Dirichlet or periodic boundary conditions. Our construction holds in any dimension d≤3 and relies on the theory of regularity structures: it yields a self-adjoint operator in L2((−L,L)d) with pure point spectrum. In d≥2, a renormalisation of the operator by means of infinite constants is required to compensate for ill-defined products involving functionals of the white noise. We also obtain left tail estimates on the distributions of the eigenvalues: in particular, for d=3 these estimates show that the eigenvalues do not have exponential moments.
Subjects / KeywordsAnderson hamiltonian; Regularity structures; White noise; Schrödinger operator
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