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On the Well-posedness of Magnetic Schrödinger Equations with Unbounded Potentials

Dorothee Frey, Siliang Weng

Abstract

We consider magnetic Schrödinger equations with sublinear magnetic potentials and subquadratic electric potentials on $\mathbb{R}^{d}$, as well as generalizations thereof. We obtain new results on the global well-posedness of the Cauchy problem with initial data in magnetic modulation spaces $M^{p}_{A}(\mathbb{R}^{d})$. Our results are achieved by approximating the solution in phase space using the magnetic Hamiltonian flow. This method includes the potentials as part of the generalized Schrödinger operator instead of treating them as perturbations, and thereby allows us to deal with unbounded potentials. For $A \equiv 0$, the space $M^{p}_{A}(\mathbb{R}^{d})$ reduces to the usual modulation space $M^{p}(\mathbb{R}^{d})$, for which relevant known results for the usual Schrödinger equation can be recovered.

On the Well-posedness of Magnetic Schrödinger Equations with Unbounded Potentials

Abstract

We consider magnetic Schrödinger equations with sublinear magnetic potentials and subquadratic electric potentials on , as well as generalizations thereof. We obtain new results on the global well-posedness of the Cauchy problem with initial data in magnetic modulation spaces . Our results are achieved by approximating the solution in phase space using the magnetic Hamiltonian flow. This method includes the potentials as part of the generalized Schrödinger operator instead of treating them as perturbations, and thereby allows us to deal with unbounded potentials. For , the space reduces to the usual modulation space , for which relevant known results for the usual Schrödinger equation can be recovered.
Paper Structure (13 sections, 20 theorems, 245 equations)

This paper contains 13 sections, 20 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Magnetic Potential and Flux
  5. Magnetic Pseudo-differential Operator
  6. The Magnetic Wavepacket Transform and Modulation Spaces
  7. Magnetic Phase Space Flow
  8. Phase space approximation
  9. Conjugating magnetic $\Psi$DOs
  10. Propagation along the phase space flow
  11. Modulation space well-posedness
  12. Integral kernel estimates
  13. Extension to the case of time-dependent magnetic fields

Key Result

Theorem 1.5

Let $T > 0$ and $1\leq p \leq \infty$. For a given magnetic field $B$ satisfying Assumption asmp:general and a real-valued symbol $h$ satisfying Assumption asmp:h, the Cauchy problem with initial value $u_{0} \in M^{2,p}_{A}(\mathbb{R}^{d})$ has a unique strong solution $u \in W^{1,1}([0,T],M_{A}^{p}(\mathbb{R}^{d}))$ with $u(t) \in M^{2,p}_{A}(\mathbb{R}^{d})$ for all $t \in [0,T]$. Moreover, we

Theorems & Definitions (41)

  • Definition 1.2
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Definition 3.1
  • Lemma 3.2
  • Definition 4.1
  • Definition 4.2
  • ...and 31 more