Semiclassical resolvent bounds in dimension two

TL;DR

This paper provides an elementary proof of resolvent bounds for semiclassical Schrödinger operators in two dimensions, showing exponential growth in the inverse parameter and linear growth near infinity, filling a gap in prior research.

Contribution

It introduces a simplified proof technique for resolvent bounds in 2D, extending previous results to include the case with Lipschitz potentials and long-range decay.

Findings

Resolvent norm grows exponentially with inverse semiclassical parameter.

Near infinity, the resolvent norm grows linearly.

The method covers cases previously unaddressed in literature.

Abstract

We give an elementary proof of weighted resolvent bounds for semiclassical Schr\"odinger operators in dimension two. We require the potential function to be Lipschitz with long range decay. The resolvent norm grows exponentially in the inverse semiclassical parameter, but near infinity it grows linearly. Our result covers the missing case from the work of Datchev.