Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral analysis of a semiclassical random walk associated to a general confining potential

Thomas Normand

Abstract

We consider a semiclassical random walk with respect to a probability measure associated to a potential with a finite number of critical points. We recover the spectral results from [1] on the corresponding operator in a more general setting and with improved accuracy. In particular we do not make any assumption on the distribution of the critical points of the potential, in the spirit of [15]. Our approach consists in adapting the ideas from [15] to the recent gaussian quasimodes framework which appears to be more robust than the usual methods, especially when dealing with non local operators.

Spectral analysis of a semiclassical random walk associated to a general confining potential

Abstract

We consider a semiclassical random walk with respect to a probability measure associated to a potential with a finite number of critical points. We recover the spectral results from [1] on the corresponding operator in a more general setting and with improved accuracy. In particular we do not make any assumption on the distribution of the critical points of the potential, in the spirit of [15]. Our approach consists in adapting the ideas from [15] to the recent gaussian quasimodes framework which appears to be more robust than the usual methods, especially when dealing with non local operators.
Paper Structure (13 sections, 20 theorems, 146 equations)

This paper contains 13 sections, 20 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Setting
  4. General labeling of the potential minima
  5. Gaussian quasimodes
  6. Orthogonality
  7. Action of the operator $P_h$
  8. Choice of $\ell^{\mathbf s,\tilde{\mathbf m}}$
  9. Solving for $\ell_0^{\mathbf s,\tilde{\mathbf m}}$
  10. Solving for $(\ell_j^{\mathbf s,\tilde{\mathbf m}})_{j\geq 1}$
  11. Construction of $\ell^{\mathbf s,\tilde{\mathbf m}}$
  12. Interaction between two wells
  13. Interaction between two quasimodes

Key Result

Theorem 1.4

Under Hypothesis hyporw, there exist $p\in \mathbb N$ and a finite set $\mathcal{A}$ both explicit as well as some positive definite matrices $(\mathcal{M}^{\alpha,j}_h)_{1\leq j\leq p\,;\, \alpha\in \mathcal{A}}$ depending on $\varrho$ from Hypothesis hyporw and admitting a classical expansion whos where $\{\hat{S}_1<\dots < \hat{S}_p\}$ are the finite values taken by the map $S$ introduced in De

Theorems & Definitions (33)

  • Definition 1.2
  • Theorem 1.4
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 3.1
  • Proposition 4.1
  • Remark 5.1
  • Lemma 5.2
  • ...and 23 more