Uniform Resolvent and Orthonormal Strichartz Estimates for Repulsive Hamiltonian

Abstract

We consider mapping properties of resolvents and the orthonormal Strichartz estimates for the Schrödinger operator. First we prove the Kato–Yajima-type uniform resolvent estimates for the repulsive Schrödinger operator with logarithmic decaying weight functions. This is new even when without perturbations. Moreover, our class of perturbations includes logarithmic decaying potentials, which is also new. The proof depends on the microlocal analysis and the Mourre theory. Then we use the Kato–Yajima estimates to prove the orthonormal Strichartz estimates. In the proof, we also use an abstract Ginibre–Velo or 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 abstract theorem is a partial extension of the recent results by Nguyen (J Funct Anal 286(1):110196, 2024) and also applies to many other Schrödinger propagators which are difficult to treat by the smooth perturbation theory, for example, local-in-time estimates for the Schrödinger operator with unbounded electromagnetic potentials, for the (k, a)-generalized Laguerre operators and global-in-time estimates for the Schrödinger operator with scaling critical magnetic potentials including the Aharonov–Bohm potentials. Finally, we also discuss mapping properties of resolvents of the repulsive Schrödinger operator on the Schwartz class. Since the principle symbol of our Hamiltonian does not belong to good symbol classes, resolvents do not preserve the Schwartz class. However, we prove that functions in the range of resolvents are Schwartz functions microlocally outside the radial sink. These results are different from related works Taira (Commun Math Phys 388(1):625–655, 2021) and Kameoka (J Spectr Theory 11(2):677–708, 2021).