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

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.

References

Milton Abramowitz (ed.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1965. Superintendent of Documents. MR 0177136
Shmuel Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 2, 151–218. MR 397194
M. Beceanu and W. Schlag, Structure formulas for wave operators, Amer. J. Math. 142 (2020), no. 3, 751–807. MR 4101331, DOI 10.1353/ajm.2020.0025
David Beltran, Jonathan Hickman, and Christopher D. Sogge, Sharp local smoothing estimates for Fourier integral operators, Geometric aspects of harmonic analysis, Springer INdAM Ser., vol. 45, Springer, Cham, [2021] ©2021, pp. 29–105. MR 4390222, DOI 10.1007/978-3-030-72058-2_{2}
Aline Bonami and Jean-Louis Clerc, Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques, Trans. Amer. Math. Soc. 183 (1973), 223–263 (French). MR 338697, DOI 10.1090/S0002-9947-1973-0338697-5
Jean-Marc Bouclet and Haruya Mizutani, Uniform resolvent and Strichartz estimates for Schrödinger equations with critical singularities, Trans. Amer. Math. Soc. 370 (2018), no. 10, 7293–7333. MR 3841849, DOI 10.1090/tran/7243
Gilles Evéquoz, Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equation in the plane, Analysis (Berlin) 37 (2017), no. 2, 55–68. MR 3657859, DOI 10.1515/anly-2016-0023
Anne Boutet de Monvel, Galina Kazantseva, and Marius Mantoiu, Some anisotropic Schrödinger operators without singular spectrum, Helv. Phys. Acta 69 (1996), no. 1, 13–25. MR 1398558
Nicolas Burq, Fabrice Planchon, John G. Stalker, and A. Shadi Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal. 203 (2003), no. 2, 519–549. MR 2003358, DOI 10.1016/S0022-1236(03)00238-6
Nicolas Burq, Fabrice Planchon, John G. Stalker, and A. Shadi Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J. 53 (2004), no. 6, 1665–1680. MR 2106340, DOI 10.1512/iumj.2004.53.2541
T. M. Dunster, On the order derivatives of Bessel functions, Constr. Approx. 46 (2017), no. 1, 47–68. MR 3668630, DOI 10.1007/s00365-016-9355-1
Javier Duoandikoetxea, Fourier analysis, Graduate Studies in Mathematics, vol. 29, American Mathematical Society, Providence, RI, 2001. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. MR 1800316, DOI 10.1090/gsm/029
Luca Fanelli, Veronica Felli, Marco A. Fontelos, and Ana Primo, Time decay of scaling critical electromagnetic Schrödinger flows, Comm. Math. Phys. 324 (2013), no. 3, 1033–1067. MR 3123544, DOI 10.1007/s00220-013-1830-y
Luca Fanelli, Veronica Felli, Marco A. Fontelos, and Ana Primo, Time decay of scaling invariant electromagnetic Schrödinger equations on the plane, Comm. Math. Phys. 337 (2015), no. 3, 1515–1533. MR 3339184, DOI 10.1007/s00220-015-2291-2
C. Gao, B. Liu, C. Miao, and Y. Xi, Improved local smoothing estimates for the wave equation in higher dimension, arXiv:2108.06870v1.
Susana Gutiérrez, Non trivial $L^q$ solutions to the Ginzburg-Landau equation, Math. Ann. 328 (2004), no. 1-2, 1–25. MR 2030368, DOI 10.1007/s00208-003-0444-7
H. Kalf, U.-W. Schmincke, J. Walter, and R. Wüst, On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., Vol. 448, Springer, Berlin, 1975, pp. 182–226. MR 0397192
Tosio Kato and S. T. Kuroda, The abstract theory of scattering, Rocky Mountain J. Math. 1 (1971), no. 1, 127–171. MR 385604, DOI 10.1216/RMJ-1971-1-1-127
Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048, DOI 10.1353/ajm.1998.0039
C. E. Kenig, A. Ruiz, and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 (1987), no. 2, 329–347. MR 894584, DOI 10.1215/S0012-7094-87-05518-9
Anatoly A. Kilbas and Megumi Saigo, $H$-transforms, Analytical Methods and Special Functions, vol. 9, Chapman & Hall/CRC, Boca Raton, FL, 2004. Theory and applications. MR 2041257, DOI 10.1201/9780203487372
R. Killip, C. Miao, M. Visan, J. Zhang, and J. Zheng, Sobolev spaces adapted to the Schrödinger operator with inverse-square potential, Math. Z. 288 (2018), no. 3-4, 1273–1298. MR 3778997, DOI 10.1007/s00209-017-1934-8
Rowan Killip, Changxing Miao, Monica Visan, Junyong Zhang, and Jiqiang Zheng, The energy-critical NLS with inverse-square potential, Discrete Contin. Dyn. Syst. 37 (2017), no. 7, 3831–3866. MR 3639442, DOI 10.3934/dcds.2017162
Rowan Killip, Jason Murphy, Monica Visan, and Jiqiang Zheng, The focusing cubic NLS with inverse-square potential in three space dimensions, Differential Integral Equations 30 (2017), no. 3-4, 161–206. MR 3611498
Rowan Killip, Monica Visan, and Xiaoyi Zhang, Riesz transforms outside a convex obstacle, Int. Math. Res. Not. IMRN 19 (2016), 5875–5921. MR 3567262, DOI 10.1093/imrn/rnv338
B. G. Korenev, Bessel functions and their applications, Analytical Methods and Special Functions, vol. 8, Taylor & Francis Group, London, 2002. Translated from the Russian by E. V. Pankratiev. MR 1963816, DOI 10.1201/b12551
S. T. Kuroda, An introduction to scattering theory, Lecture Notes Series, vol. 51, Aarhus Universitet, Matematisk Institut, Aarhus, 1978. MR 528757
Jing Lu, Changxing Miao, and Jason Murphy, Scattering in $H^1$ for the intercritical NLS with an inverse-square potential, J. Differential Equations 264 (2018), no. 5, 3174–3211. MR 3741387, DOI 10.1016/j.jde.2017.11.015
Changxing Miao, Jason Murphy, and Jiqiang Zheng, The energy-critical nonlinear wave equation with an inverse-square potential, Ann. Inst. H. Poincaré C Anal. Non Linéaire 37 (2020), no. 2, 417–456. MR 4072805, DOI 10.1016/j.anihpc.2019.09.004
Haruya Mizutani, Remarks on endpoint Strichartz estimates for Schrödinger equations with the critical inverse-square potential, J. Differential Equations 263 (2017), no. 7, 3832–3853. MR 3670038, DOI 10.1016/j.jde.2017.05.006
Haruya Mizutani, Uniform Sobolev estimates for Schrödinger operators with scaling-critical potentials and applications, Anal. PDE 13 (2020), no. 5, 1333–1369. MR 4149064, DOI 10.2140/apde.2020.13.1333
Haruya Mizutani, Scattering theory in homogeneous Sobolev spaces for Schrödinger and wave equations with rough potentials, J. Math. Phys. 61 (2020), no. 9, 091505, 21. MR 4146725, DOI 10.1063/5.0019682
Haruya Mizutani, Junyong Zhang, and Jiqiang Zheng, Uniform resolvent estimates for Schrödinger operator with an inverse-square potential, J. Funct. Anal. 278 (2020), no. 4, 108350, 29. MR 4044742, DOI 10.1016/j.jfa.2019.108350
F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.
Fabrice Planchon, John G. Stalker, and A. Shadi Tahvildar-Zadeh, $L^p$ estimates for the wave equation with the inverse-square potential, Discrete Contin. Dyn. Syst. 9 (2003), no. 2, 427–442. MR 1952384, DOI 10.3934/dcds.2003.9.427
Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0493420
Christopher D. Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991), no. 2, 349–376. MR 1098614, DOI 10.1007/BF01245080
Christopher D. Sogge, Fourier integrals in classical analysis, 2nd ed., Cambridge Tracts in Mathematics, vol. 210, Cambridge University Press, Cambridge, 2017. MR 3645429, DOI 10.1017/9781316341186
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 512086
F. G. Tricomi and A. Erdélyi, The asymptotic expansion of a ratio of gamma functions, Pacific J. Math. 1 (1951), 133–142. MR 43948, DOI 10.2140/pjm.1951.1.133
Juan Luis Vazquez and Enrike Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal. 173 (2000), no. 1, 103–153. MR 1760280, DOI 10.1006/jfan.1999.3556
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
Kenji Yajima, The $W^{k,p}$-continuity of wave operators for Schrödinger operators, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 4, 94–98. MR 1222831
Kenji Yajima, The $W^{k,p}$-continuity of wave operators for Schrödinger operators. II. Positive potentials in even dimensions $m\ge 4$, Spectral and scattering theory (Sanda, 1992) Lecture Notes in Pure and Appl. Math., vol. 161, Dekker, New York, 1994, pp. 287–300. MR 1291648
Kenji Yajima, The $W^{k,p}$-continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan 47 (1995), no. 3, 551–581. MR 1331331, DOI 10.2969/jmsj/04730551
Kenji Yajima, The $W^{k,p}$-continuity of wave operators for Schrödinger operators. III. Even-dimensional cases $m\geq 4$, J. Math. Sci. Univ. Tokyo 2 (1995), no. 2, 311–346. MR 1366561
Kenji Yajima, $L^p$-boundedness of wave operators for two-dimensional Schrödinger operators, Comm. Math. Phys. 208 (1999), no. 1, 125–152. MR 1729881, DOI 10.1007/s002200050751
K. Yajima, The $L^p$ boundedness of wave operators for Schrödinger operators with threshold singularities. I. The odd dimensional case, J. Math. Sci. Univ. Tokyo 13 (2006), no. 1, 43–93. MR 2223681
Junyong Zhang and Jiqiang Zheng, Scattering theory for nonlinear Schrödinger equations with inverse-square potential, J. Funct. Anal. 267 (2014), no. 8, 2907–2932. MR 3255478, DOI 10.1016/j.jfa.2014.08.012
Jiqiang Zheng, Focusing NLS with inverse square potential, J. Math. Phys. 59 (2018), no. 11, 111502, 14. MR 3872306, DOI 10.1063/1.5054167