Smoothing estimates for the Schrodinger equation with unbounded potentials

TL;DR

This paper establishes local smoothing estimates for the Schrödinger equation with unbounded polynomially growing potentials, using gauge-invariant assumptions and multiplier methods without pseudodifferential tools.

Contribution

It provides the first smoothing estimates for magnetic Schrödinger equations with unbounded potentials under gauge-invariant conditions using elementary multiplier techniques.

Findings

Proves local smoothing estimates for unbounded magnetic Schrödinger equations.

Assumptions involve only first two derivatives of the magnetic field.

No pseudodifferential calculus needed for the proof.

Abstract

We prove a local in time smoothing estimate for a magnetic Schrodinger equation with coefficients growing polynomially at spatial infinity. The assumptions on the magnetic field are gauge invariant and involve only the first two derivatives. The proof is based on the multiplier method and no pseudofferential techniques are required.