Decay estimates for Schrödinger's equation with magnetic potentials in three dimensions

TL;DR

This work proves that the Schrödinger evolution with a magnetic potential in \(\mathbb{R}^3\), governed by\n$H=-\Delta+i(A\nabla+\nabla A)+V$, exhibits the same dispersive decay as the free equation on the absolutely continuous spectrum, namely a bound of the form $\|e^{itH}P_{ac}f\|_{L^\infty}\lesssim |t|^{-3/2}\|f\|_{L^1}$ under short-range hypotheses and a regular zero-energy threshold. The authors develop a robust operator-algebra framework built on Kato-type spaces and the algebroid $\mathcal{U}(X,Y)$, analyze the free and perturbed resolvents via a resolvent identity, and control the perturbation by bilinear estimates for terms like $R_0(\lambda^2)\nabla A R_0(\lambda^2)\nabla A^#$. A key step is the use of Wiener’s theorem to invert a perturbation series $(I-T(\lambda))^{-1}$ in a space tailored to the magnetic-plus-potential structure, which yields uniform bounds for the perturbed resolvent $R(\lambda^2)$ and its derivative. Consequently, the paper extends scalar-potential dispersive theory to magnetic cases, provides sharp time-decay for the Schrödinger flow, and notes analogous implications for wave-type equations and threshold spectral phenomena (no embedded eigenvalues under the assumptions, finitely many negative eigenvalues in general).

Abstract

In this paper we prove that Schrödinger's equation with a Hamiltonian of the form $H=-Δ+i(A \nabla + \nabla A) + V$, which includes a magnetic potential $A$, has the same dispersive and solution decay properties as the free Schrödinger equation. In particular, we prove $L^1 \to L^\infty$ decay and some related estimates for the wave equation. The potentials $A$ and $V$ are short-range and $A$ has four derivatives, but they can be arbitrarily large. All results hold in three space dimensions.

Theorems & Definitions (37)

• Theorem 1
• Definition 1
• Definition 2
• Theorem 2: Wiener
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof