Abstract
We give conditions on radial nonnegative weights \(W_1\) and \(W_2\) on \({\Bbb R}^n\), for which the a priori inequality
holds with constant independent of \(k \in {\Bbb R}\). Here \(\Delta_S\) is the Laplace-Beltrami operator on the sphere \({\Bbb S}^{n−1}\). Due to the relation between \((-\Delta_S)^{1/2}\) and the tangential component of the gradient, \(\nabla_{\tau}\), we obtain some "Morawetz-type" estimates for \(\nabla_{\tau} u\) on \({\Bbb R}^n; n \geq 3\). As a consequence we establish some new estimates for the free Schrödinger propagator \(e^{it\Delta}\), which may be viewed as certain refinements of the \(\Delta\)-(super)smoothness estimates of Kato and Yajima. These results, in turn, lead to the well-posedness of the initial value problem for certain time dependent first order spherical perturbations of the \(n \geq 3\) dimensional Schrödinger equation.
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Barceló, J., Bennett, J. & Ruiz, A. Spherical Perturbations of Schrödinger Equations. J Fourier Anal Appl 12, 269–290 (2006). https://doi.org/10.1007/s00041-006-6003-3
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DOI: https://doi.org/10.1007/s00041-006-6003-3