Uniform resolvent and orthonormal Strichartz estimates for repulsive Hamiltonian

TL;DR

This paper establishes uniform resolvent and orthonormal Strichartz estimates for the Schr"odinger operator with repulsive potentials, extending to complex cases like electromagnetic and magnetic potentials, using microlocal analysis and Mourre theory.

Contribution

It proves a Keel-Tao type theorem for orthonormal Strichartz estimates applicable to difficult Schr"odinger propagators and introduces new uniform resolvent estimates for repulsive Hamiltonians.

Findings

Dispersive estimates imply orthonormal Strichartz estimates for various Schr"odinger propagators.

Established Kato-Yajima type uniform resolvent estimates with logarithmic weights.

Extended mapping properties of resolvents to Schwartz and Lebesgue spaces.

Abstract

We consider the uniform resolvent and orthonormal Strichartz estimates for the Schr\"odinger operator. First we prove the Keel-Tao type theorem for the orthonormal Strichartz estimates, which means that the dispersive estimates yield the orthonormal Strichartz estimates for strongly continuous unitary groups. This result applies to many Schr\"odinger propagators which are difficult to treat by the smooth perturbation theory, for example, local-in-time estimates for the Schr\"odinger operator with unbounded electromagnetic potentials, the $(k, a)$-generalized Laguerre operators and global-in-time estimates for the Schr\"odinger operator with scaling critical magnetic potentials including the Aharonov-Bohm potentials. Next we observe mapping properties of resolvents for the repulsive Hamiltonian and apply to the orthonormal Strichartz estimates. We prove the Kato-Yajima type uniform resolvent estimates with logarithmic decaying weight functions. This is new even when without perturbations. The proof is dependent on the microlocal analysis and the Mourre theory. We also discuss mapping properties on the Schwartz class and the Lebesgue space.