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June  2019, 8(2): 447-471. doi: 10.3934/eect.2019022

Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods

Toshiyuki Suzuki

Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama city, Kanagawa, JAPAN

Received  February 2018 Revised  September 2018 Published  June 2019 Early access  March 2019

We solve the scattering problems for nonlinear Schrödinger equations with an inverse-square potential by applying the energy methods. The methods are optimized to the abstract semilinear Schrödinger evolution equations with nonautonomous terms.

Keywords: Scattering problems, semilinear Schrödinger equations, Hartree equations, energy methods, inverse-square potentials, nonautonomous system.
Mathematics Subject Classification: Primary: 35Q55, 35Q40; Secondary: 81Q15.
Citation: Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations & Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022
References:
[1]

L. Baudouin, O. Kavian and J.-P. Puel, Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control, J. Differential Equations, 216 (2005), 188-222.  doi: 10.1016/j.jde.2005.04.006.  Google Scholar

[2]

N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549.  doi: 10.1016/S0022-1236(03)00238-6.  Google Scholar

[3]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, 2003. doi: 10.1090/cln/010.  Google Scholar

[4]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[5]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.  doi: 10.1007/BF02099529.  Google Scholar

[6]

J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension $n \ge 2$, Comm. Math. Phys., 151 (1993), 619-645.  doi: 10.1007/BF02097031.  Google Scholar

[7]

J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. Ⅰ, Rev. Math. Phys., 12 (2000), 361-429.  doi: 10.1142/S0129055X00000137.  Google Scholar

[8]

J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. Ⅱ, Ann. Henri Poincaré, 1 (2000), 753-800.  doi: 10.1007/PL00001014.  Google Scholar

[9]

N. Hayashi and T. Ozawa, Scattering theory in the weighted $L^{2}(\mathbb{R}^{n})$ spaces for some Schrödinger equations, Ann. Inst. Henri Poincaré, 48 (1988), 17-37.   Google Scholar

[10]

N. Hayashi and Y. Tsutsumi, Scattering theory for Hartree type equations, Ann. Inst. Henri Poincaré, 46 (1987), 187-213.   Google Scholar

[11]

J. Lu, C. Miao and J. Murphy, Scattering in $H^{1}$ for the intercritical NLS with an inverse-square potential, J. Differ. Equ., 264 (2018), 3174-3211.  doi: 10.1016/j.jde.2017.11.015.  Google Scholar

[12]

H. Mizutani, Remarks on endpoint Strichartz estimates for Schrödinger equations with the critical inverse-square potential, J. Differential Equations, 263 (2017), 3832-3853.  doi: 10.1016/j.jde.2017.05.006.  Google Scholar

[13]

K. Nakanishi, Modified wave operators for the Hartree equation with data, image and convergence in the same space. Ⅱ, Ann. Henri Poincaré, 3 (2002), 503-535.  doi: 10.1007/s00023-002-8626-5.  Google Scholar

[14]

K. Nakanishi, Modified wave operators for the Hartree equation with data, image and convergence in the same space, Commun. Pure Appl. Anal., 1 (2002), 237-252.  doi: 10.3934/cpaa.2002.1.237.  Google Scholar

[15]

N. Okazawa, T. Suzuki and T. Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials, Appl. Anal., 91 (2012), 1605-1629.  doi: 10.1080/00036811.2011.631914.  Google Scholar

[16]

N. Okazawa, T. Suzuki and T. Yokota, Energy methods for abstract nonlinear Schrödinger equations, Evol. Equ. Control Theory, 1 (2012), 337-354.  doi: 10.3934/eect.2012.1.337.  Google Scholar

[17]

N. Okazawa, T. Yokota and K. Yoshii, Remarks on linear Schrödinger evolution equations with Coulomb potential with moving center, SUT J. Math., 46 (2010), 155-176.   Google Scholar

[18]

N. Okazawa and K. Yoshii, Linear Schrödinger evolution equations with moving Coulomb singularities, J. Differential Equations, 254 (2013), 2964-2999.  doi: 10.1016/j.jde.2013.01.017.  Google Scholar

[19]

V. Pierfelice, Weighted Strichartz estimates for the Schrödinger and wave equations on Damek-Ricci spaces, Math. Z., 260 (2008), 377-392.   Google Scholar

[20]

T. Suzuki, Energy methods for Hartree type equation with inverse-square potentials, Evol. Equ. Control Theory, 2 (2013), 531-542.  doi: 10.3934/eect.2013.2.531.  Google Scholar

[21]

T. Suzuki, Blowup of nonlinear Schrödinger equations with inverse-square potentials, Differ. Equ. Appl., 6 (2014), 309-333.  doi: 10.7153/dea-06-17.  Google Scholar

[22]

T. Suzuki, Solvability of nonlinear Schrödinger equations with some critical singular potential via generalized Hardy-Rellich inequalities, Funkcial. Ekvac., 59 (2016), 1-34.  doi: 10.1619/fesi.59.1.  Google Scholar

[23]

T. Suzuki, Scattering theory for Hartree equations with inverse-square potentials, Appl. Anal., 96 (2017), 2032-2043.  doi: 10.1080/00036811.2016.1200720.  Google Scholar

[24]

T. Suzuki, Virial identities for nonlinear Schrödinger equations with an inverse-square potential of critical coefficient, Differ. Equ. Appl., 9 (2017), 327-352.  doi: 10.7153/dea-2017-09-24.  Google Scholar

[25]

K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 110 (1987), 415-426.  doi: 10.1007/BF01212420.  Google Scholar

[26]

J. Zhang and J. Zheng, Scattering theory for nonlinear Schrödinger equations with inverse-square potential, J. Funct. Anal., 267 (2014), 2907-2932.  doi: 10.1016/j.jfa.2014.08.012.  Google Scholar

show all references

References:
[1]

L. Baudouin, O. Kavian and J.-P. Puel, Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control, J. Differential Equations, 216 (2005), 188-222.  doi: 10.1016/j.jde.2005.04.006.  Google Scholar

[2]

N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549.  doi: 10.1016/S0022-1236(03)00238-6.  Google Scholar

[3]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, 2003. doi: 10.1090/cln/010.  Google Scholar

[4]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[5]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.  doi: 10.1007/BF02099529.  Google Scholar

[6]

J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension $n \ge 2$, Comm. Math. Phys., 151 (1993), 619-645.  doi: 10.1007/BF02097031.  Google Scholar

[7]

J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. Ⅰ, Rev. Math. Phys., 12 (2000), 361-429.  doi: 10.1142/S0129055X00000137.  Google Scholar

[8]

J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. Ⅱ, Ann. Henri Poincaré, 1 (2000), 753-800.  doi: 10.1007/PL00001014.  Google Scholar

[9]

N. Hayashi and T. Ozawa, Scattering theory in the weighted $L^{2}(\mathbb{R}^{n})$ spaces for some Schrödinger equations, Ann. Inst. Henri Poincaré, 48 (1988), 17-37.   Google Scholar

[10]

N. Hayashi and Y. Tsutsumi, Scattering theory for Hartree type equations, Ann. Inst. Henri Poincaré, 46 (1987), 187-213.   Google Scholar

[11]

J. Lu, C. Miao and J. Murphy, Scattering in $H^{1}$ for the intercritical NLS with an inverse-square potential, J. Differ. Equ., 264 (2018), 3174-3211.  doi: 10.1016/j.jde.2017.11.015.  Google Scholar

[12]

H. Mizutani, Remarks on endpoint Strichartz estimates for Schrödinger equations with the critical inverse-square potential, J. Differential Equations, 263 (2017), 3832-3853.  doi: 10.1016/j.jde.2017.05.006.  Google Scholar

[13]

K. Nakanishi, Modified wave operators for the Hartree equation with data, image and convergence in the same space. Ⅱ, Ann. Henri Poincaré, 3 (2002), 503-535.  doi: 10.1007/s00023-002-8626-5.  Google Scholar

[14]

K. Nakanishi, Modified wave operators for the Hartree equation with data, image and convergence in the same space, Commun. Pure Appl. Anal., 1 (2002), 237-252.  doi: 10.3934/cpaa.2002.1.237.  Google Scholar

[15]

N. Okazawa, T. Suzuki and T. Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials, Appl. Anal., 91 (2012), 1605-1629.  doi: 10.1080/00036811.2011.631914.  Google Scholar

[16]

N. Okazawa, T. Suzuki and T. Yokota, Energy methods for abstract nonlinear Schrödinger equations, Evol. Equ. Control Theory, 1 (2012), 337-354.  doi: 10.3934/eect.2012.1.337.  Google Scholar

[17]

N. Okazawa, T. Yokota and K. Yoshii, Remarks on linear Schrödinger evolution equations with Coulomb potential with moving center, SUT J. Math., 46 (2010), 155-176.   Google Scholar

[18]

N. Okazawa and K. Yoshii, Linear Schrödinger evolution equations with moving Coulomb singularities, J. Differential Equations, 254 (2013), 2964-2999.  doi: 10.1016/j.jde.2013.01.017.  Google Scholar

[19]

V. Pierfelice, Weighted Strichartz estimates for the Schrödinger and wave equations on Damek-Ricci spaces, Math. Z., 260 (2008), 377-392.   Google Scholar

[20]

T. Suzuki, Energy methods for Hartree type equation with inverse-square potentials, Evol. Equ. Control Theory, 2 (2013), 531-542.  doi: 10.3934/eect.2013.2.531.  Google Scholar

[21]

T. Suzuki, Blowup of nonlinear Schrödinger equations with inverse-square potentials, Differ. Equ. Appl., 6 (2014), 309-333.  doi: 10.7153/dea-06-17.  Google Scholar

[22]

T. Suzuki, Solvability of nonlinear Schrödinger equations with some critical singular potential via generalized Hardy-Rellich inequalities, Funkcial. Ekvac., 59 (2016), 1-34.  doi: 10.1619/fesi.59.1.  Google Scholar

[23]

T. Suzuki, Scattering theory for Hartree equations with inverse-square potentials, Appl. Anal., 96 (2017), 2032-2043.  doi: 10.1080/00036811.2016.1200720.  Google Scholar

[24]

T. Suzuki, Virial identities for nonlinear Schrödinger equations with an inverse-square potential of critical coefficient, Differ. Equ. Appl., 9 (2017), 327-352.  doi: 10.7153/dea-2017-09-24.  Google Scholar

[25]

K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 110 (1987), 415-426.  doi: 10.1007/BF01212420.  Google Scholar

[26]

J. Zhang and J. Zheng, Scattering theory for nonlinear Schrödinger equations with inverse-square potential, J. Funct. Anal., 267 (2014), 2907-2932.  doi: 10.1016/j.jfa.2014.08.012.  Google Scholar

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