Laplaciens De Graphes Infinis I-Graphes Métriquement Complets

Abstract

A la mémoire de mon père Pr. Dr. Ing. Bèchir Torki (1931-2009 Résumé: On introduit le Laplacien ∆ ω,c d'un graphe G localement fini pondéréà la fois sur les sommets et sur les arêtes, ainsi que la notion d'opérateur de Schrödinger ∆ 1,a + W . Pour les graphesà poids constants sur les sommets, onétend un résultat de Wojciechowski pour le Laplacien et un résultat de Dodziuk pour les opérateurs de Schrödinger concernant le caractère essentiellement auto-adjoint. Le résultat principal de ce travailétablit que pour les graphes pondérésà valence bornée et métriquement complets, le Laplacien ∆ ω,c est essentiellement auto-adjoint, et il en va de même pour l' opérateur ∆ 1,a + W pourvu que la forme quadratique associée soit minorée. La preuve fait appelà la construction d'une fonction harmonique strictement positive qui permet d'écrire l'opérateur de Schrödinger ∆ 1,a + W comme un Laplacieǹ a poids ∆ ω,cà transformation unitaire près. * Math Subject Calssification (2000) : 05C63, 05C50, 05C12, 35J10, 47B25. † Mots clés : graphe infini, Laplacien de graphe pondéré, opŕateur de Schrödinger, essentiellement auto-adjoint.

A Sears-type self-adjointness result for discrete magnetic Schrödinger operators

“…This result of [3] was extended in [11] to semi-bounded below operator H| C c (V ) in ℓ 2 w (V ), where H is as in (6). For a study of essential self-adjointness of Schrödinger operators (without magnetic field) on a metrically non-complete graph, see [4].…”

“…For the case σ ≡ 1, see [2,3]. Let W : V → R, and consider a Schrödinger-type expression Hu := ∆ σ u + Wu.…”

“…For ∆ as in (11) with w ≡ 1, the essential self-adjointness of ∆| C c (V ) was proven in [16] (see also [17]). Additionally, the paper [3] established the essential self-adjointness of (∆ + W )| C c (V ) , where ∆ is as in (11) with w ≡ c 0 and W ≥ −c 1 , where c 0 > 0 and c 1 ∈ R are constants. The results of [14][15][16] on the essential self-adjointness of ∆ and the result of [3] on the essential self-adjointness of (∆ [18,19].…”

“…Additionally, the paper [3] established the essential self-adjointness of (∆ + W )| C c (V ) , where ∆ is as in (11) with w ≡ c 0 and W ≥ −c 1 , where c 0 > 0 and c 1 ∈ R are constants. The results of [14][15][16] on the essential self-adjointness of ∆ and the result of [3] on the essential self-adjointness of (∆ [18,19]. The authors of [18,19] worked on discrete sets, a more general context than locally finite graphs, and gave sufficient conditions for the essential self-adjointness of generators of Dirichlet forms.…”

“…In the context of a graph of bounded degree, the paper [3] contains an important link between the essential selfadjointness of (∆ + W )| C c (V ) , where ∆ is as in (11) with w ≡ 1, and completeness of the weighted metric d 1,a;1 in (8) above; namely, if (G, d 1,a;1 ) is complete and if (∆ + W )| C c (V ) is semi-bounded below, then (∆ + W )| C c (V ) is essentially selfadjoint in the space ℓ 2 w (V ) with w ≡ 1. This result of [3] was extended in [11] to semi-bounded below operator H| C c (V ) in ℓ 2 w (V ), where H is as in (6).…”