Resolvent estimates for the magnetic Hamiltonian with singular vector potentials and applications

TL;DR

This paper develops resolvent estimates for magnetic Hamiltonians with singular vector potentials, enabling analysis of resonances and smoothing effects in Schrödinger solutions, advancing understanding of such quantum systems.

Contribution

It introduces a method for analytically continuing the resolvent and deriving estimates near the real axis for Hamiltonians with singular potentials, which was previously challenging.

Findings

Resolved the resolvent analytically near the positive real axis.

Derived asymptotic locations of resonances.

Established a local smoothing estimate for Schrödinger solutions.

Abstract

For the magnetic Hamiltonian with singular vector potentials, we analytically continue the resolvent to a logarithmic neighborhood of the positive real axis and prove resolvent estimates there. As applications, we obtain asymptotic locations of resonances and a local smoothing estimate for solutions of the corresponding Schr\"odinger equation.