Strichartz estimates for quadratic repulsive potentials

TL;DR

This paper establishes Strichartz estimates for quantum particles under quadratic repulsive potentials, demonstrating decay properties and extending results to perturbed Hamiltonians with slowly decaying potentials.

Contribution

It proves Strichartz estimates for both free and perturbed quadratic repulsive Hamiltonians, including cases with slowly decaying potentials, expanding understanding of dispersive properties in such systems.

Findings

Strichartz estimates hold for all admissible pairs under free repulsive Hamiltonian.

Estimates extend to perturbed Hamiltonians with potentials decaying as |x|^{- ext{delta}}.

Fast decay of solutions due to exponential acceleration by the potential.

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.