Quantitative limiting absorption principle in the semiclassical limit

TL;DR

This paper provides an elementary proof of resolvent bounds for long-range semiclassical Schrödinger operators, showing exponential growth globally and linear growth near infinity, while also relaxing regularity assumptions on the potential.

Contribution

It introduces a simplified proof of Burq's resolvent bounds and extends the results to less regular potentials in the semiclassical limit.

Findings

Resolvent norm grows exponentially in the inverse semiclassical parameter

Near infinity, the resolvent norm grows linearly

Regularity assumptions on the potential are weakened

Abstract

We give an elementary proof of Burq's resolvent bounds for long range semiclassical Schroedinger operators. Globally, the resolvent norm grows exponentially in the inverse semiclassical parameter, and near infinity it grows linearly. We also weaken the regularity assumptions on the potential.