Uniform Sobolev estimates for Schr\"odinger operators with scaling-critical potentials and applications

TL;DR

This paper establishes uniform Sobolev estimates for Schr"odinger operators with critical potentials, leading to new results in Strichartz estimates, spectral multipliers, and eigenvalue bounds, broadening understanding of these operators without repulsive conditions.

Contribution

It introduces uniform Sobolev estimates for Schr"odinger operators with scaling-critical potentials without requiring repulsive conditions, enabling various applications.

Findings

Proved uniform Sobolev estimates for critical potentials

Derived global-in-time Strichartz estimates, including non-admissible ones

Established spectral multiplier theorems and eigenvalue bounds for complex potentials

Abstract

We prove uniform Sobolev estimates for the resolvent of Schr\"odinger operators with large scaling-critical potentials without any repulsive condition. As applications, global-in-time Strichartz estimates including some non-admissible retarded estimates, a H\"ormander type spectral multiplier theorem, and Keller type eigenvalue bounds with complex-valued potentials are also obtained.