The 𝑊^{𝑠,𝑝}-boundedness of stationary wave operators for the Schrödinger operator with inverse-square potential

Miao, Changxing; Su, Xiaoyan; Zheng, Jiqiang

doi:10.1090/tran/8823

The $W^{s,p}$-boundedness of stationary wave operators for the Schrödinger operator with inverse-square potential
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by Changxing Miao, Xiaoyan Su and Jiqiang Zheng;

Trans. Amer. Math. Soc. 376 (2023), 1739-1797

DOI: https://doi.org/10.1090/tran/8823

Published electronically: December 8, 2022

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Abstract:

In this paper, we investigate the $W^{s,p}$-boundedness for stationary wave operators of the Schrödinger operator with inverse-square potential \begin{equation*} \mathcal L_a=-\Delta +\tfrac {a}{|x|^2}, \quad a\geq -\tfrac {(d-2)^2}{4}, \end{equation*} in dimension $d\geq 2$. We construct the stationary wave operators in terms of integrals of Bessel functions and spherical harmonics, and prove that they are $W^{s,p}$-bounded for certain $p$ and $s$ which depend on $a$. As corollaries, we solve some open problems associated with the operator $\mathcal L_a$, which include the dispersive estimates and the local smoothing estimates in dimension $d\geq 2$. We also generalize some known results such as the uniform Sobolev inequalities, the equivalence of Sobolev norms and the Mikhlin multiplier theorem, to a larger range of indices. These results are important in the description of linear and nonlinear dynamics for dispersive equations with inverse-square potential.

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