Dunkl harmonic oscillator and Witten's perturbation on strata

Abstract

The main goal of this work is to use Witten's perturbation method to prove a version of Morse inequalities for the minimum and maximum ideal boundary conditions of the de Rham complex on strata, endowed with adapted metrics, where compact Thom-Mather strati cations are considered. For that purpose, we study rst eigenfunction estimates and embedding results for the Dunkl harmonic oscillator on the line, which are generalized to other related operators on R+. The study of these operators is the key ingredient in our local analysis of the Witten's perturbation. Thus this thesis has two main parts. The rst part is devoted to the study of eigenfunction estimates and embedding results for the Dunkl harmonic oscillator and related operators. The second part deals with the Witten's perturbation on strata, where the rst part is used. This work is published in the preprints [1, 2]. Let us introduce those chapters separately and state their main results.