# The bottom of the spectrum of time-changed processes and the maximum principle of Schr\"{o}dinger operators

## TL;DR

This paper establishes a criterion linking the maximum principle of Schrödinger operators to the spectrum of time-changed processes, providing insights into their properties and implications for the Liouville property.

## Contribution

It introduces a necessary and sufficient condition for the maximum principle of Schrödinger operators based on the bottom of the spectrum of associated time-changed processes.

## Key findings

- Characterization of the maximum principle via spectral conditions
- Sufficient conditions for the Liouville property
- Connection between spectral bottom and operator properties

## Abstract

We give a necessary and sufficient condition for the maximum principle of Schr\"{o}dinger operators in terms of the bottom of the spectrum of time-changed processes. As a corollary, we obtain a sufficient condition for the Liouville property of Schr\"{o}dinger operators.