Uniform resolvent and orthonormal Strichartz estimates for repulsive Hamiltonian

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.