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absence de résonance près du réel pour l’opérateur de Schrödinger

Nicolas Burq (1997/1998)

Séminaire Équations aux dérivées partielles

On donne dans cet exposé des bornes inférieures universelles, en limite semiclassique, de la hauteur des résonances de forme associées aux opérateurs de Schrödinger à l’extérieur d’obstacles avec des conditions au bord de Dirichlet ou de Neumann et des potentiels analytiquement dilatables et tendant vers 0 à l’infini. Ces bornes inférieures sont exponentiellement petites par rapport à la constante de Planck.

Absolute continuity for Jacobi matrices with power-like weights

Wojciech Motyka (2007)

Colloquium Mathematicae

This work deals with a class of Jacobi matrices with power-like weights. The main theme is spectral analysis of matrices with zero diagonal and weights λ : = n α ( 1 + Δ ) where α ∈ (0,1]. Asymptotic formulas for generalized eigenvectors are given and absolute continuity of the matrices considered is proved. The last section is devoted to spectral analysis of Jacobi matrices with qₙ = n + 1 + (-1)ⁿ and λ = ( q n - 1 q ) .

Absolutely continuous linear operators on Köthe-Bochner spaces

(2011)

Banach Center Publications

Let E be a Banach function space over a finite and atomless measure space (Ω,Σ,μ) and let ( X , | | · | | X ) and ( Y , | | · | | Y ) be real Banach spaces. A linear operator T acting from the Köthe-Bochner space E(X) to Y is said to be absolutely continuous if | | T ( 1 A f ) | | Y 0 whenever μ(Aₙ) → 0, (Aₙ) ⊂ Σ. In this paper we examine absolutely continuous operators from E(X) to Y. Moreover, we establish relationships between different classes of linear operators from E(X) to Y.

Absolutely continuous spectrum and scattering in the surface Maryland model

François Bentosela, Philippe Briet, Leonid Pastur (2001)

Journées équations aux dérivées partielles

We study the discrete Schrödinger operator H in 𝐙 d with the surface quasi periodic potential V ( x ) = g δ ( x 1 ) tan π ( α · x 2 + ω ) , where x = ( x 1 , x 2 ) , x 1 𝐙 d 1 , x 2 𝐙 d 2 , α 𝐑 d 2 , ω [ 0 , 1 ) . We first discuss a proof of the pure absolute continuity of the spectrum of H on the interval [ - d , d ] (the spectrum of the discrete laplacian) in the case where the components of α are rationally independent. Then we show that in this case the generalized eigenfunctions have the form of the “volume” waves, i.e. of the sum of the incident plane wave and reflected from the hyper-plane 𝐙 d 1 waves, the form...

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