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

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  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.

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. BaudouinO. 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. BurqF. PlanchonJ. 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. LuC. 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. OkazawaT. 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. OkazawaT. 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. OkazawaT. 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. BaudouinO. 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. BurqF. PlanchonJ. 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. LuC. 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. OkazawaT. 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. OkazawaT. 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. OkazawaT. 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

[1]

Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2947-2969. doi: 10.3934/dcds.2020392

[2]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298

[3]

Kazuhiro Kurata, Yuki Osada. Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021100

[4]

Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024

[5]

Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems & Imaging, 2021, 15 (3) : 499-517. doi: 10.3934/ipi.2021002

[6]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

[7]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450

[8]

Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216

[9]

Flank D. M. Bezerra, Jacson Simsen, Mariza Stefanello Simsen. Convergence of quasilinear parabolic equations to semilinear equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3823-3834. doi: 10.3934/dcdsb.2020258

[10]

Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021013

[11]

Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021047

[12]

Fangyi Qin, Jun Wang, Jing Yang. Infinitely many positive solutions for Schrödinger-poisson systems with nonsymmetry potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021054

[13]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

[14]

Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3817-3836. doi: 10.3934/dcds.2021018

[15]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[16]

Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021038

[17]

Abdeslem Hafid Bentbib, Smahane El-Halouy, El Mostafa Sadek. Extended Krylov subspace methods for solving Sylvester and Stein tensor equations. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021026

[18]

Guodong Wang, Bijun Zuo. Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021078

[19]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447

[20]

Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (123)
  • HTML views (487)
  • Cited by (2)

Other articles
by authors

[Back to Top]