## Convergence of a class of Schrödinger equations.(English)Zbl 1440.35287

Summary: We set up the selection conditions for time series $$\{t_k\}_{k = 1}^\infty$$ which converge to 0 as $$k \to \infty$$ and for which the solutions of a class of generalized Schrödinger equations pointwise converge almost everywhere to their initial data in $$H^s (\mathbb{R}^n)$$ for $$s > 0$$.

### MSC:

 35Q41 Time-dependent Schrödinger equations and Dirac equations 37M10 Time series analysis of dynamical systems

### Keywords:

Schrödinger equation; convergence; Sobolev spaces
Full Text:

### References:

 [1] J. Bourgain, “Some new estimates on oscillatory integrals”, pp. 83-112 in Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), edited by C. Fefferman et al., Princeton Math. Ser. 42, Princeton Univ. Press, 1995. [2] J. Bourgain, “On the Schrödinger maximal function in higher dimension”, Proc. Steklov Inst. Math. 280 (2013), 46-60. · Zbl 1287.65041 [3] J. Bourgain, “A note on the Schrödinger maximal function”, J. Anal. Math. 130 (2016), 393-396. · Zbl 1361.35151 [4] L. Carleson, “Some analytic problems related to statistical mechanics”, pp. 5-45 in Euclidean harmonic analysis (Univ. Maryland, College Park, MD, 1979), edited by J. J. Benedetto, Lecture Notes in Math. 779, Springer, 1980. [5] C.-H. Cho and H. Ko, “A note on maximal estimates of generalized Schrödinger equation”, 2018. [6] C.-H. Cho, S. Lee, and A. Vargas, “Problems on pointwise convergence of solutions to the Schrödinger equation”, J. Fourier Anal. Appl. 18:5 (2012), 972-994. · Zbl 1264.35184 [7] B. E. J. Dahlberg and C. E. Kenig, “A note on the almost everywhere behavior of solutions to the Schrödinger equation”, pp. 205-209 in Harmonic analysis (Minneapolis, MN, 1981), edited by F. Ricci and G. Weiss, Lecture Notes in Math. 908, Springer, 1982. [8] Y. Ding and Y. Niu, “Weighted maximal estimates along curve associated with dispersive equations”, Anal. Appl. $$($$ Singap.$$) 15$$:2 (2017), 225-240. · Zbl 1362.42027 [9] X. Du and R. Zhang, “Sharp $$L^2$$ estimates of the Schrödinger maximal function in higher dimensions”, Ann. of Math. $$(2) 189$$:3 (2019), 837-861. · Zbl 1433.42010 [10] X. Du, L. Guth, and X. Li, “A sharp Schrödinger maximal estimate in $$\mathbb R^2$$”, Ann. of Math. $$(2) 186$$:2 (2017), 607-640. · Zbl 1378.42011 [11] X. Du, L. Guth, X. Li, and R. Zhang, “Pointwise convergence of Schrödinger solutions and multilinear refined Strichartz estimates”, Forum Math. Sigma 6 (2018), art. id. e14. · Zbl 1395.42063 [12] X. Du, J. Kim, H. Wang, and R. Zhang, “Lower bounds for estimates of the Schrödinger maximal function”, 2019. [13] S. Lee, “On pointwise convergence of the solutions to Schrödinger equations in $$\mathbb R^2$$”, Int. Math. Res. Not. 2006 (2006), art. id. 32597. · Zbl 1131.35306 [14] W. Li and H. Wang, “Pointwise convergence of solutions to the Schrödinger equation along a class of curves”, 2018. [15] C. Miao, J. Yang, and J. Zheng, “An improved maximal inequality for 2D fractional order Schrödinger operators”, Studia Math. 230:2 (2015), 121-165. · Zbl 1343.42027 [16] A. Moyua, A. Vargas, and L. Vega, “Schrödinger maximal function and restriction properties of the Fourier transform”, Internat. Math. Res. Notices 1996:16 (1996), 793-815. · Zbl 0868.35024 [17] P. Sjögren and P. Sjölin, “Convergence properties for the time-dependent Schrödinger equation”, Ann. Acad. Sci. Fenn. Ser. A I Math. 14:1 (1989), 13-25. · Zbl 0629.35055 [18] P. Sjölin, “Regularity of solutions to the Schrödinger equation”, Duke Math. J. 55:3 (1987), 699-715. · Zbl 0631.42010 [19] P. Sjölin, “Two theorems on convergence of Schrödinger means”, J. Fourier Anal. Appl. 25:4 (2019), 1708-1716. · Zbl 1417.42026 [20] T. Tao and A. Vargas, “A bilinear approach to cone multipliers, II: Applications”, Geom. Funct. Anal. 10:1 (2000), 216-258. · Zbl 0949.42013 [21] L. Vega, El multiplicador de Schrödinger: la función maximal y los operadores de restricción, Ph.D. thesis, Universidad Autónoma de Madrid, 1988, http://hdl.handle.net/10486/131206. [22] L.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.