# Exponential lower resolvent bounds far away from trapped sets

## TL;DR

This paper constructs examples of semiclassical Schrödinger operators with exponentially large resolvent norms far from trapped sets, revealing threshold radii where resolvent and wave decay behaviors change.

## Contribution

It provides explicit radial examples demonstrating exponential resolvent growth far from trapped sets and analyzes the change in behavior at a critical radius.

## Key findings

- Exponential lower bounds on resolvent norms far from trapped sets
- Identification of a threshold radius affecting resolvent behavior
- Application to wave equations showing change in decay properties

## Abstract

We give examples of semiclassical Schr\"odinger operators with exponentially large cutoff resolvent norms, even when the supports of the cutoff and potential are very far apart. The examples are radial, which allows us to analyze the resolvent kernel in detail using ordinary differential equation techniques. In particular, we identify a threshold spatial radius where the resolvent behavior changes. We apply these results to wave equations with radial wavespeed, identifying a corresponding threshold radius at which wave decay properties change.