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Wave packet decomposition for Schrodinger evolution with rough potential and generic value of parameter

Sergey A. Denisov

TL;DR

This work develops a wave packet framework to analyze Schrödinger evolution with rough time-dependent potentials, introducing a central one-collision operator $Q$ and a $k$-dependent wave packet decomposition that expresses free evolution as a superposition of tubes with slowly varying envelopes. By partitioning space-time into characteristic cubes and sparsifying the potential, the authors isolate leading resonant interactions and prove a quantitative bound on $Q$ that mitigates the potential’s impact for generic spectral parameters, extending perturbative results below the naive threshold. They further classify resonant versus nonresonant contributions, show nonresonant terms are negligible via phase analysis, and quantify the resonance configurations that can distort propagation. Importantly, the paper provides explicit constructions showing resonances can occur for $\gamma<1$, demonstrating that nonresonant propagation is not universal in this regime and highlighting the delicate balance between dispersive transport and rough perturbations. The techniques advance understanding of transport in dispersive PDEs with rough or structured perturbations and offer a framework potentially applicable to Anderson-type settings and related perturbative analyses.

Abstract

We develop the wave packet decomposition to study the Schrodinger evolution with rough potential. As an application, we obtain the improved bound on the wave propagation for the generic value of a parameter.

Wave packet decomposition for Schrodinger evolution with rough potential and generic value of parameter

TL;DR

This work develops a wave packet framework to analyze Schrödinger evolution with rough time-dependent potentials, introducing a central one-collision operator and a -dependent wave packet decomposition that expresses free evolution as a superposition of tubes with slowly varying envelopes. By partitioning space-time into characteristic cubes and sparsifying the potential, the authors isolate leading resonant interactions and prove a quantitative bound on that mitigates the potential’s impact for generic spectral parameters, extending perturbative results below the naive threshold. They further classify resonant versus nonresonant contributions, show nonresonant terms are negligible via phase analysis, and quantify the resonance configurations that can distort propagation. Importantly, the paper provides explicit constructions showing resonances can occur for , demonstrating that nonresonant propagation is not universal in this regime and highlighting the delicate balance between dispersive transport and rough perturbations. The techniques advance understanding of transport in dispersive PDEs with rough or structured perturbations and offer a framework potentially applicable to Anderson-type settings and related perturbative analyses.

Abstract

We develop the wave packet decomposition to study the Schrodinger evolution with rough potential. As an application, we obtain the improved bound on the wave propagation for the generic value of a parameter.
Paper Structure (10 sections, 10 theorems, 136 equations)

This paper contains 10 sections, 10 theorems, 136 equations.

Table of Contents

  1. Introduction.
  2. The main estimate and the wave packet decomposition.
  3. $k$-dependent wave packet decomposition.
  4. Interaction of wave packets with $V$ on characteristic cubes.
  5. Contribution from each characteristic cube $B_{p,q}$.
  6. Sparsifying $V$.
  7. Contribution from all of $V$.
  8. Resonant terms and proofs of the main results.
  9. Creation of the resonances.
  10. Appendix.

Key Result

Lemma 1.1

Suppose $0\leqslant t_1\leqslant t\leqslant t_2\leqslant 2\pi T$ and $\|V\|_{\infty}\lesssim T^{-\gamma}$. Then, and the operator norm of $Err$ allows the estimate $\|Err\|\lesssim T^{-2\gamma}(t_2-t_1)^2\,.$

Theorems & Definitions (16)

  • Lemma 1.1
  • proof
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Lemma 2.1
  • proof
  • Theorem 2.2: The main estimate for "one-collision operator"
  • Lemma 3.1
  • proof
  • ...and 6 more