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Local Decay Estimates

Avy Soffer, Xiaoxu Wu

Abstract

We give a proof of Local Decay Estimates for Schrödinger type equations, which is based on the knowledge of Asymptotic Completeness (AC). This approach extends to time dependent potential perturbations, as it does not rely on Resolvent Estimates or related methods. Global in time Strichartz estimates follow for quasi-periodic time-dependent potentials from our results.

Local Decay Estimates

Abstract

We give a proof of Local Decay Estimates for Schrödinger type equations, which is based on the knowledge of Asymptotic Completeness (AC). This approach extends to time dependent potential perturbations, as it does not rely on Resolvent Estimates or related methods. Global in time Strichartz estimates follow for quasi-periodic time-dependent potentials from our results.
Paper Structure (23 sections, 40 theorems, 357 equations)

This paper contains 23 sections, 40 theorems, 357 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. The time-dependent result
  4. The time-independent result
  5. Applications
  6. Outline of the proof
  7. Challenges of the Analysis of Time-Dependent Hamiltonians
  8. Previous methods
  9. Connecting Quasi-Periodic Time Dependence to General Time Dependence
  10. Phase Space Decompositions
  11. Incoming/outg oing waves
  12. Quasi-periodic evolution operators
  13. The time-independent problem
  14. Compactness of $\mathcal{C}_c$
  15. Decomposition of the Operator $\mathcal{C}_c$
  16. ...and 8 more sections

Key Result

Theorem 1.1

If Assumption aspV2 is satisfied, then with $p_0=2$, goal: 1 holds true for all $p_0\leq p<\infty$, when $n\geq 8$.

Theorems & Definitions (86)

  • Definition 1.1: Quasi-periodic functions
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Theorem 1.4
  • Definition 2.1: Incoming/outgoing waves
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 76 more