# On fractional multi-singular Schr\"odinger operators: positivity and localization of binding

## TL;DR

This paper studies the positivity and localization of binding for fractional Schr"odinger operators with complex potentials, providing criteria linking spectrum positivity to supersolutions and analyzing pole configurations.

## Contribution

It introduces a new criterion connecting spectrum positivity with supersolutions and characterizes pole arrangements that guarantee operator positivity.

## Key findings

- Established necessary and sufficient conditions for positivity based on pole configurations.
- Developed a criterion linking spectrum positivity to the existence of positive supersolutions.
- Analyzed localization of binding for nonlocal Schr"odinger operators with multipolar potentials.

## Abstract

In this work we investigate positivity properties of nonlocal Schr\"odinger type operators, driven by the fractional Laplacian, with multipolar, critical, and locally homogeneous potentials. On one hand, we develop a criterion that links the positivity of the spectrum of such operators with the existence of certain positive supersolutions, while, on the other hand, we study the localization of binding for this kind of potentials. Combining these two tools and performing an inductive procedure on the number of poles, we establish necessary and sufficient conditions for the existence of a configuration of poles that ensures the positivity of the corresponding Schr\"odinger operator.