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Wave front set of solutions to Schrödinger equations with time-dependent magnetic fields

Fumihito Abe, Ryo Muramatsu

TL;DR

The paper addresses the microlocal analysis of Schrödinger evolution with time-dependent magnetic fields by characterizing the wave front set of solutions in terms of the initial data using the wave packet transform developed by Kato, Kobayashi, and Ito. It derives a phase-space representation of the solution, proves a robust equivalence between WF absence and decay of appropriately scaled wave packets along classical trajectories, and extends the framework to a short-time scalar potential perturbation. A key application shows that the fundamental solution has no singularities for magnetic fields decaying at infinity, unveiling a microlocal smoothing effect despite long-range, time-dependent magnetic fields. These results advance the understanding of propagation of singularities in magnetic Schrödinger operators and enable analysis under spatially growing, time-dependent vector potentials, with potential relevance to quantum dynamics in decaying fields.

Abstract

In this paper, we determine the wave front set of solutions to the Schrödinger equation with time-dependent magnetic fields. We considered time-dependent and `not so small' magnetic fields through the method using the wave packet transform established by K. Kato, M. Kobayashi and S. Ito. Furthermore, we checked that the fundamental solution of the Schrödinger equation in a spatially decaying magnetic field has no singularities as a consequence of our result.

Wave front set of solutions to Schrödinger equations with time-dependent magnetic fields

TL;DR

The paper addresses the microlocal analysis of Schrödinger evolution with time-dependent magnetic fields by characterizing the wave front set of solutions in terms of the initial data using the wave packet transform developed by Kato, Kobayashi, and Ito. It derives a phase-space representation of the solution, proves a robust equivalence between WF absence and decay of appropriately scaled wave packets along classical trajectories, and extends the framework to a short-time scalar potential perturbation. A key application shows that the fundamental solution has no singularities for magnetic fields decaying at infinity, unveiling a microlocal smoothing effect despite long-range, time-dependent magnetic fields. These results advance the understanding of propagation of singularities in magnetic Schrödinger operators and enable analysis under spatially growing, time-dependent vector potentials, with potential relevance to quantum dynamics in decaying fields.

Abstract

In this paper, we determine the wave front set of solutions to the Schrödinger equation with time-dependent magnetic fields. We considered time-dependent and `not so small' magnetic fields through the method using the wave packet transform established by K. Kato, M. Kobayashi and S. Ito. Furthermore, we checked that the fundamental solution of the Schrödinger equation in a spatially decaying magnetic field has no singularities as a consequence of our result.
Paper Structure (8 sections, 10 theorems, 61 equations)

This paper contains 8 sections, 10 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Transform of equation \ref{['eq1']}
  4. Characterization of wave front set
  5. Key lemmas
  6. Proof of Theorem \ref{['mt1']}
  7. Proof of Corollaries
  8. Appendix

Key Result

Theorem 1.1

Suppose $\bm{a}(t,x)$ satisfies Assumption A2. Let $b=\min\left(\frac{1}{8}, \frac{1-\rho}{8}\right)$. Then for any $u_0(x)\in L^2(\mathbb{R}^n)$, $\varphi_0(x)\in {\mathcal{S}}(\mathbb{R}^n)\setminus \{0\}$, the solution $u(t,x)\in C(\mathbb{R};L^2(\mathbb{R}^n))$ of eq1 and $t_0\in\mathbb{R}$, the

Theorems & Definitions (21)

  • Definition 1.2: Wave packet transform
  • Definition 1.3: wave front set
  • Theorem 1.1
  • Corollary 1.4
  • Corollary 1.5: Absence of singularities of the fundamental solution
  • Definition 2.1: Inverse wave packet transform
  • Lemma 2.2
  • proof
  • Proposition 2.3: K. Kato, M. Kobayashi and S. Ito KKI3
  • Remark 2.4
  • ...and 11 more