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Dispersive estimates for Schrodinger operators in dimensions one and three

M. Goldberg, W. Schlag

TL;DR

This work proves sharp L^1→L^∞ dispersive bounds for Schrödinger evolutions e^{itH}P_ac in dimensions 1 and 3 under minimal decay assumptions on the potential V. The authors employ a hybrid approach: large-energy behavior is controlled by a finite Born-series resolvent expansion with Kato-type norms, while low-energy behavior is analyzed via zero-energy resolvent expansions (Jost theory in 1D and Jensen–Kato style in 3D), with careful handling of zero-energy resonances. In 1D, the bound |t|^{−1/2} holds for V∈L^1_1 with no zero-energy resonance, and extends to V∈L^1_2 regardless of resonance; in 3D, under β>3 decay and absence of zero-energy eigenvalue or resonance, the bound |t|^{−3/2} is obtained. The paper also discusses resonance phenomena, optimality considerations, and how the methods might extend to other dimensions.

Abstract

We prove L^1 --> L^\infty estimates for linear Schroedinger equations in dimensions one and three. The potentials are only required to satisfy some mild decay assumptions. No regularity on the potentials is assumed.

Dispersive estimates for Schrodinger operators in dimensions one and three

TL;DR

This work proves sharp L^1→L^∞ dispersive bounds for Schrödinger evolutions e^{itH}P_ac in dimensions 1 and 3 under minimal decay assumptions on the potential V. The authors employ a hybrid approach: large-energy behavior is controlled by a finite Born-series resolvent expansion with Kato-type norms, while low-energy behavior is analyzed via zero-energy resolvent expansions (Jost theory in 1D and Jensen–Kato style in 3D), with careful handling of zero-energy resonances. In 1D, the bound |t|^{−1/2} holds for V∈L^1_1 with no zero-energy resonance, and extends to V∈L^1_2 regardless of resonance; in 3D, under β>3 decay and absence of zero-energy eigenvalue or resonance, the bound |t|^{−3/2} is obtained. The paper also discusses resonance phenomena, optimality considerations, and how the methods might extend to other dimensions.

Abstract

We prove L^1 --> L^\infty estimates for linear Schroedinger equations in dimensions one and three. The potentials are only required to satisfy some mild decay assumptions. No regularity on the potentials is assumed.

Paper Structure

This paper contains 5 sections, 15 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. The one-dimensional case
  3. The three-dimensional case
  4. Large energies
  5. Low energies

Key Result

Theorem 1

Let $V\in L^1_1({\mathbb R})$, i.e., $\int_{-\infty}^\infty |V(x)|(1+|x|)\,dx<\infty$, and assume that there is no resonance at zero energy. Then for all $t$ where $H=-\frac{d^2}{dx^2}+V$. The conclusion holds for all $V \in L^1_2({\mathbb R})$, i.e., $\int_{-\infty}^\infty |V(x)|(1+|x|)^{2}\,dx<\infty$, whether or not there is a resonance at zero energy.

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • proof
  • proof : Proof of Lemma \ref{['lem:low']}
  • Definition 6
  • Lemma 7
  • ...and 19 more