  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.
