Théorie de Mourre et opérateurs de Schrödinger : De nouvelles classes d'opérateurs conjugués

Abstract : In this thesis, we are interested in the study of the essential spectrum of Schrödinger operators and more particulary in the obtention of a Limiting Absorption Principle for these operators. This Limiting Absorption Principle consists on the existence of a limit for the resolvent operator when the spectral parameter is near the essential spectrum and permits to know some properties about the group generated by the Schrödinger Hamiltonian we study. A technique to prove this Limiting Absorption Principle is to use the Mourre theory. This theory needs to use an other operator called the conjugate operator. When we want to apply the Mourre theory to Schrödinger operators, we usually used a conjugate operatornamed the generator of dilations. This operator implies some conditions of decay on the derivatives of the potentials which can be a problem in certain cases. In this thesis, we will apply the Mourre theory with other types of conjugate operators wich, for some of them, does not imply any conditions on the derivatives of the potential.In a first part, we will be interested in Schrödinger operators on the euclidian space. We will show a Limiting Absorption Principle at positive energy, a Limiting Absorption principle at zero energy and the absence of eigenvalue embedded in the essential spectrum. In a second part, we will be interested in Schrödinger operators on wave guides for which we will prove a Limiting Absorption Principle far thresholds and near thresholds.
Document type :
Theses
Complete list of metadatas

Cited literature [59 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-02283002
Contributor : Abes Star <>
Submitted on : Tuesday, September 10, 2019 - 1:36:36 PM
Last modification on : Friday, September 13, 2019 - 2:29:09 PM

File

68828_MARTIN_2018_diffusion.pd...
Version validated by the jury (STAR)

Identifiers

  • HAL Id : tel-02283002, version 1

Collections

Citation

Alexandre Martin. Théorie de Mourre et opérateurs de Schrödinger : De nouvelles classes d'opérateurs conjugués. Mathématiques générales [math.GM]. Université de Cergy Pontoise, 2018. Français. ⟨NNT : 2018CERG0978⟩. ⟨tel-02283002⟩

Share

Metrics

Record views

41

Files downloads

8