Abstract
For the one-dimensional Schrödinger equation, we obtain sharp maximal-in-time and maximal-in-space estimates for systems of orthonormal initial data. The maximal-in-time estimates generalize a classical result of Kenig–Ponce–Vega and allow us to obtain pointwise convergence results associated with systems of infinitely many fermions. The maximal-in-space estimates simultaneously address an endpoint problem raised by Frank–Sabin in their work on Strichartz estimates for orthonormal systems of data, and provide a path toward proving our maximal-in-time estimates.
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Notes
If \(\beta < 2\) and \((\nu _j^*)_j\) is the sequence \((|\nu _j|)_j\) permuted in a decreasing order, we have \( \Vert \nu \Vert _{\beta '} \lesssim (\sum _{j \ge 1} (\nu _j^*)^{\beta '} j^{\beta '/2} \cdot j^{-\beta '/2})^{1/\beta '} \lesssim \sup _{j \ge 1} j^{1/2} \nu _j^* = \Vert \nu \Vert _{\ell ^{2,\infty }} \) and therefore, by duality, \(\ell ^{\beta } \subseteq \ell ^{2,1}\).
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Acknowledgements
This work was supported by JSPS Kakenhi grant numbers 18KK0073 and 19H01796 (Bez), Korean Research Foundation Grant No. NRF-2018R1A2B2006298 (Lee), and Grant-in-Aid for JSPS Research Fellow No. 17J01766 (Nakamura).
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Bez, N., Lee, S. & Nakamura, S. Maximal estimates for the Schrödinger equation with orthonormal initial data. Sel. Math. New Ser. 26, 52 (2020). https://doi.org/10.1007/s00029-020-00582-6
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DOI: https://doi.org/10.1007/s00029-020-00582-6