Dispersive estimates for Schroedinger operators: A survey

TL;DR

The article surveys dispersive estimates for Schrödinger evolutions $e^{itH}P_c$ with decaying real potentials across dimensions and in time-dependent settings. It synthesizes perturbative and spectral techniques, including Born series, resolvent expansions near zero energy, and the limiting absorption principle, to explain how zero-energy phenomena (eigenvalues and resonances) affect decay rates. The survey highlights dimension-specific results: in $d\ge3$ perturbative and large-potential regimes; in $d=1$ with sharp weighted decay when zero is not a resonance; in $d=2$ a first $L^1\to L^\infty$ bound with $|t|^{-1}$ decay under a regular-zero condition, via threshold-resolvent expansions. It also covers time-dependent potentials, Floquet theory, and charge-transfer models, connecting dispersive estimates to scattering theory and nonlinear stability questions in PDEs.

Abstract

We present some old and new results on dispersive estimates for Schroedinger equations.

Theorems & Definitions (10)

• Lemma 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• Theorem 3.1
• proof
• Theorem 4.1
• Lemma 4.2