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Local decay estimates for the bi-Laplacian Nonautonomous Schrödinger equation

Jiayan Wu, Ting Zhang, Ruze Zhou

Abstract

In this paper, we establish local decay estimates for the bi-Laplacian Schrödinger equation with time-dependent (in particular, quasi-periodic) potentials in spatial dimension $n\ge14$. Moreover, under stronger spectral regularity hypotheses, the same result can be extended to dimension $n\ge9$. Our approach, based on asymptotic completeness and the existence of the channel wave operator, departs from standard resolvent-based methods. In addition, global-in-time Strichartz estimates are derived from the local decay estimates.

Local decay estimates for the bi-Laplacian Nonautonomous Schrödinger equation

Abstract

In this paper, we establish local decay estimates for the bi-Laplacian Schrödinger equation with time-dependent (in particular, quasi-periodic) potentials in spatial dimension . Moreover, under stronger spectral regularity hypotheses, the same result can be extended to dimension . Our approach, based on asymptotic completeness and the existence of the channel wave operator, departs from standard resolvent-based methods. In addition, global-in-time Strichartz estimates are derived from the local decay estimates.

Paper Structure

This paper contains 25 sections, 36 theorems, 343 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminary and main results
  4. Applications
  5. Notations
  6. Strategies
  7. Difficulties in the problems and recent advances in dispersion Theory
  8. Phase Space Decompositions
  9. Incoming/Outgoing Waves
  10. Quasi-Periodic Evolution Operators
  11. Preprocessing of the operator (C(t))
  12. Compactness of (C(t)) and its decomposition
  13. A Smallness Analysis on (C r(t))
  14. Elaborate estimate for each part of the operator
  15. Momentum-Position Weight Absorption Mechanism of the operator (C M(t))
  16. ...and 10 more sections

Key Result

Theorem 1.1

If Assumptions asp:1 and asp:2 hold, then, for all $\eta>\frac{5}{2}$, holds for $n\geq 14$.

Theorems & Definitions (77)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 1.7
  • Definition 2.1: Cutoff on the forward/backward propagation set
  • Lemma 2.2
  • proof
  • ...and 67 more