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The homogeneous repulsive potentials accelerate a quantum particle and the velocity of the particle increases exponentially in t; this phenomenon yields the fast decaying dispersive estimates and hence we consider the Strichartz estimates associated with this phenomenon. First, we consider the free repulsive Hamiltonian and prove that the Strichartz estimates hold for every admissible pairs (q, r), which satisfy 1/q+n/(2r) ≥ n/4 with q, r ≥ 2. Second, we consider the perturbed repulsive Hamiltonian with the slowly decaying potential such that |V (x)| ≤ C(1 + |x|) −δ for some δ > 0, and prove the Strichartz estimate with the same admissible pairs for the free case.

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In this paper, we give a characterization of the ranges of the wave operators for Schrödinger equations with time-dependent "short-range" potentials by using wave packet transform, which is different from the one in Kitada-Yajima [9]. We also give an alternative proof of the existence of the wave operators for time-dependent potentials, which has been firstly proved by D. R. Yafaev [14].