Abstract
We obtain approximation bounds for products of quasimodes for the Laplace–Beltrami operator on compact Riemannian manifolds of all dimensions without boundary. We approximate the products of quasimodes uv by a low-degree vector space
Acknowledgements
The research was carried out while the author was visiting Johns Hopkins University supervised by Professor C. D. Sogge. And the author would like to express her deep gratitude to Professor C. D. Sogge, for bringing this research topic to her attention, and also for the valuable guidance, helpful suggestions and comments he provided.
References
[1] M. Blair, Y. Sire and C. D. Sogge, Quasimode, eigenfunction and spectral projection bounds for Schrodinger operators on manifolds with critically singular potentials, preprint (2019), https://arxiv.org/abs/1904.09665. 10.1007/s12220-019-00287-zSearch in Google Scholar
[2] N. Burq, P. Gérard and N. Tzvetkov, Multilinear estimates for the Laplace spectral projectors on compact manifolds, C. R. Math. Acad. Sci. Paris 338 (2004), no. 5, 359–364. 10.1016/j.crma.2003.12.015Search in Google Scholar
[3] N. Burq, P. Gérard and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math. 159 (2005), no. 1, 187–223. 10.1007/s00222-004-0388-xSearch in Google Scholar
[4] N. Burq, P. Gérard and N. Tzvetkov, Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations, Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), no. 2, 255–301. 10.1016/j.ansens.2004.11.003Search in Google Scholar
[5] J. E. Colliander, J.-M. Delort, C. E. Kenig and G. Staffilani, Bilinear estimates and applications to 2D NLS, Trans. Amer. Math. Soc. 353 (2001), no. 8, 3307–3325. 10.1090/S0002-9947-01-02760-XSearch in Google Scholar
[6]
Z. Guo, X. Han and M. Tacy,
[7] H. Hirayama and S. Kinoshita, Sharp bilinear estimates and its application to a system of quadratic derivative nonlinear Schrödinger equations, Nonlinear Anal. 178 (2019), 205–226. 10.1016/j.na.2018.07.013Search in Google Scholar
[8]
H. Koch, D. Tataru and M. Zworski,
Semiclassical
[9] J. Lu, C. D. Sogge and S. Steinerberger, Approximating pointwise products of Laplacian eigenfunctions, J. Funct. Anal. 277 (2019), no. 9, 3271–3282. 10.1016/j.jfa.2019.05.025Search in Google Scholar
[10] J. F. Lu and S. Steinerberger, On pointwise products of elliptic eigenfunctions, preprint (2018), https://arxiv.org/abs/1810.01024v2. Search in Google Scholar
[11] C. D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J. 53 (1986), no. 1, 43–65. 10.1215/S0012-7094-86-05303-2Search in Google Scholar
[12]
C. D. Sogge,
Concerning the
[13] C. D. Sogge, Fourier Integrals in Classical Analysis, 2nd ed., Cambridge Tracts in Math. 210, Cambridge University Press, Cambridge, 2017. 10.1017/9781316341186Search in Google Scholar
[14]
C. D. Sogge and S. Zelditch,
A note on
[15] H. Takaoka, Bilinear Strichartz estimates and applications to the cubic nonlinear Schrödinger equation in two space dimensions, Hokkaido Math. J. 37 (2008), no. 4, 861–870. 10.14492/hokmj/1249046373Search in Google Scholar
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