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Pointwise decay for radial solutions of the Schrödinger equation with a repulsive Coulomb potential (2309.01313v2)
Published 4 Sep 2023 in math.AP
Abstract: We study the long-time behavior of solutions to the Schr\"odinger equation with a repulsive Coulomb potential on $\mathbb{R}3$ for spherically symmetric initial data. Our approach involves computing the distorted Fourier transform of the action of the associated Hamiltonian $H=-\Delta+\frac{q}{|x|}$ on radial data $f$, which allows us to explicitly write the evolution $e{itH}f$. A comprehensive analysis of the kernel is then used to establish that, for large times, $|e{i t H}f|{L{\infty}} \leq C t{-\frac{3}{2}}|f|{L1}$. Our analysis of the distorted Fourier transform is expected to have applications to other long-range repulsive problems.