Strichartz estimates for quadratic repulsive potentials

Abstract

Quadratic repulsive potentials $- \tau ^2 |x| ^2$ accelerate the quantum particle, increasing the velocity of the particle exponentially in $t$; this phenomenon yields fast decaying dispersive estimates. In this study, 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 pair $(q,r)$, which satisfies $1/q +n/(2r) \geq n/4$ with $q$, $r \geq 2$. Second, we consider the perturbed repulsive Hamiltonian with a slowly decaying potential, such that $|V(x)| \leq C(1+|x|)^{-\delta}$ for some $\delta >0$, and prove that the Strichartz estimate holds with the same admissible pairs for repulsive-admissible pairs.