# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2021002

## Inverse $N$-body scattering with the time-dependent hartree-fock approximation

 Department of Sciences, Okayama University of Science, 1-1 Ridaicho, Kita-ku, Okayama-shi 700-0005, Japan

Received  May 2020 Revised  September 2020 Published  December 2020

Fund Project: The first author is supported by JSPS KAKENHI Grant Number 19K03617

We consider an inverse $N$-body scattering problem of determining two potentials—an external potential acting on all particles and a pair interaction potential—from the scattering particles. This paper finds that the time-dependent Hartree-Fock approximation for a three-dimensional inverse $N$-body scattering in quantum mechanics enables us to recover the two potentials from the scattering states with high-velocity initial states. The main ingredient of mathematical analysis in this paper is based on the asymptotic analysis of the scattering operator defined in terms of a scattering solution to the Hartree-Fock equation at high energies. We show that the leading part of the asymptotic expansion of the scattering operator uniquely reconstructs the Fourier transform of the pair interaction, and the second term of the expansion uniquely reconstructs the $X$-ray transform of the external potential.

Citation: Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems & Imaging, doi: 10.3934/ipi.2021002
##### References:
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Phys., 36 (1995), 3902-3921.  doi: 10.1063/1.530937.  Google Scholar [6] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969.  Google Scholar [7] N. Hayashi and P. I. Naumkin, Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, Amer. J. Math, 120 (1998), 369-389.  doi: 10.1353/ajm.1998.0011.  Google Scholar [8] N. Hayashi and T. Ozawa, Scattering theory in the weighted $l^{2} (\mathbb{R}^n)$ spaces for some schrödinger equations, Annales de l'IHP Physique Théorique, 48 (1988), 17-37.   Google Scholar [9] A. Ishida, Inverse scattering in the Stark effect, Inverse Problems, 35 (2019), 105010, 20pp. doi: 10.1088/1361-6420/ab2fec.  Google Scholar [10] H. Isozaki, On the existence of solutions of time-dependent Hartree-Fock equations, Publ. Res. Inst. Math. Sci., 19 (1983), 107-115.  doi: 10.2977/prims/1195182978.  Google Scholar [11] R. Kress, Linear Integral Equations, vol. 82 of Applied Mathematical Sciences, 2nd edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0559-3.  Google Scholar [12] J. C. Lemm and J. Uhlig, Hartree-fock approximation for inverse many-body problems, Physical Review Letters, 84 (2000), 4517-4520.  doi: 10.1103/PhysRevLett.84.4517.  Google Scholar [13] C. Lubich, From quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. doi: 10.4171/067.  Google Scholar [14] K. Mochizuki, On small data scattering with cubic convolution nonlinearity, J. Math. Soc. Japan, 41 (1989), 143-160.  doi: 10.2969/jmsj/04110143.  Google Scholar [15] P. M. Morse and H. Feshbach, Methods of Theoretical Physics, 2 volumes, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953.  Google Scholar [16] R. G. Novikov, On inverse scattering for the $N$-body Schrödinger equation, J. Funct. Anal., 159 (1998), 492-536.  doi: 10.1006/jfan.1998.3324.  Google Scholar [17] A. G. Ramm and A. I. Katsevich, The Radon Transform and Local Tomography, CRC Press, Boca Raton, FL, 1996.   Google Scholar [18] W. Strauss, Nonlinear scattering theory, in Scattering Theory in Mathematical Physics (eds. L. J. A. and M. J.-P.), vol. 9 of Nato Advanced Study Institutes Series C, D. Reidel, 1974, 53–78. doi: 10.1007/978-94-010-2147-0_3.  Google Scholar [19] T. Takiguchi, An inverse problem for free channel scattering, in Inverse Problems and Related Topics (Kobe, 1998), vol. 419 of Chapman & Hall/CRC Res. Notes Math., Chapman & Hall/CRC, Boca Raton, FL, 2000,165–179.  Google Scholar [20] G. Uhlmann and A. Vasy, Low-energy inverse problems in three-body scattering, Inverse Problems, 18 (2002), 719-736.  doi: 10.1088/0266-5611/18/3/313.  Google Scholar [21] G. Uhlmann and A. 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Wang, On the uniqueness of inverse scattering for $N$-body systems, Inverse Problems, 10 (1994), 765-784.  doi: 10.1088/0266-5611/10/3/017.  Google Scholar [27] X. P. Wang, High energy asymptotics for n-body scattering matrices with arbitrary channels, Annales de l'IHP Physique Théorique, 65 (1996), 81–108.  Google Scholar [28] M. Watanabe, Inverse scattering for the nonlinear Schrödinger equation with cubic convolution nonlinearity, Tokyo J. Math., 24 (2001), 59-67.  doi: 10.3836/tjm/1255958311.  Google Scholar [29] M. Watanabe, Inverse scattering problem for time dependent Hartree-Fock equations in the three-body case, J. Math. Phys., 48 (2007), 053510, 9pp. doi: 10.1063/1.2732171.  Google Scholar [30] M. Watanabe, A remark on inverse scattering for time dependent hartree equations, in Journal of Physics: Conference Series, 73 (2007), 012025. doi: 10.1088/1742-6596/73/1/012025.  Google Scholar [31] M. Watanabe, Time-dependent method for non-linear Schrödinger equations in inverse scattering problems, J. Math. Anal. Appl., 459 (2018), 932-944.  doi: 10.1016/j.jmaa.2017.11.012.  Google Scholar [32] M. Watanabe, Time-dependent methods in inverse scattering problems for the Hartree-Fock equation, J. Math. Phys., 60 (2019), 091504, 19pp. doi: 10.1063/1.5090924.  Google Scholar [33] R. Weder, Multidimensional inverse scattering in an electric field, J. Funct. Anal., 139 (1996), 441-465.  doi: 10.1006/jfan.1996.0092.  Google Scholar [34] R. Weder, Inverse scattering for $N$-body systems with time-dependent potentials, in Inverse Problems of Wave Propagation and Diffraction (Aix-les-Bains, 1996), vol. 486 of Lecture Notes in Phys., Springer, Berlin, 1997, 27–46. doi: 10.1007/BFb0105758.  Google Scholar [35] R. Weder, Inverse scattering for the nonlinear Schrödinger equation, Comm. Partial Differential Equations, 22 (1997), 2089-2103.  doi: 10.1080/03605309708821332.  Google Scholar [36] R. Weder, Multidimensional inverse problems in perturbed stratified media, J. Differential Equations, 152 (1999), 191-239.  doi: 10.1006/jdeq.1998.3509.  Google Scholar [37] R. Weder, The $W_{k, p}$-continuity of the Schrödinger wave operators on the line, Comm. Math. Phys., 208 (1999), 507-520.  doi: 10.1007/s002200050767.  Google Scholar [38] R. Weder, Inverse scattering on the line for the nonlinear Klein-Gordon equation with a potential, J. Math. Anal. Appl., 252 (2000), 102-123.  doi: 10.1006/jmaa.2000.6954.  Google Scholar [39] R. Weder, $L^p$-$L^{\dot p}$ estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential, J. Funct. Anal., 170 (2000), 37-68.  doi: 10.1006/jfan.1999.3507.  Google Scholar [40] R. Weder, Uniqueness of inverse scattering for the nonlinear Schrödinger equation and reconstruction of the potential and the nonlinearity, in Mathematical and Numerical Aspects of Wave Propagation (Santiago de Compostela, 2000), SIAM, Philadelphia, PA, 2000,631–634.  Google Scholar [41] R. Weder, Inverse scattering for the non-linear Schrödinger equation: reconstruction of the potential and the non-linearity, Math. Methods Appl. Sci., 24 (2001), 245-254.  doi: 10.1002/mma.216.  Google Scholar [42] R. Weder, Inverse scattering for the nonlinear Schrödinger equation. II. Reconstruction of the potential and the nonlinearity in the multidimensional case, Proc. Amer. Math. Soc., 129 (2001), 3637-3645.  doi: 10.1090/S0002-9939-01-06016-6.  Google Scholar [43] R. Weder, Multidimensional inverse scattering for the nonlinear Klein-Gordon equation with a potential, J. Differential Equations, 184 (2002), 62-77.  doi: 10.1006/jdeq.2001.4133.  Google Scholar [44] R. Weder, The $L^p$-$L^{p'}$ estimate for the Schrödinger equation on the half-line, J. Math. Anal. Appl., 281 (2003), 233-243.  doi: 10.1016/S0022-247X(03)00093-3.  Google Scholar [45] K. 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), 311-346.   Google Scholar

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##### References:
 [1] T. Adachi, Y. Fujiwara and A. Ishida, On multidimensional inverse scattering in time-dependent electric fields, Inverse Problems, 29 (2013), 085012, 24pp. doi: 10.1088/0266-5611/29/8/085012.  Google Scholar [2] T. Adachi, T. Kamada, M. Kazuno and K. Toratani, On multidimensional inverse scattering in an external electric field asymptotically zero in time, Inverse Problems, 27 (2011), 065006, 17pp. doi: 10.1088/0266-5611/27/6/065006.  Google Scholar [3] T. Adachi and K. Maehara, On multidimensional inverse scattering for Stark Hamiltonians, J. Math. Phys., 48 (2007), 042101, 12pp. doi: 10.1063/1.2713077.  Google Scholar [4] T. Cazenave, Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar [5] V. Enss and R. Weder, The geometrical approach to multidimensional inverse scattering, J. Math. Phys., 36 (1995), 3902-3921.  doi: 10.1063/1.530937.  Google Scholar [6] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969.  Google Scholar [7] N. Hayashi and P. I. Naumkin, Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, Amer. J. Math, 120 (1998), 369-389.  doi: 10.1353/ajm.1998.0011.  Google Scholar [8] N. Hayashi and T. Ozawa, Scattering theory in the weighted $l^{2} (\mathbb{R}^n)$ spaces for some schrödinger equations, Annales de l'IHP Physique Théorique, 48 (1988), 17-37.   Google Scholar [9] A. Ishida, Inverse scattering in the Stark effect, Inverse Problems, 35 (2019), 105010, 20pp. doi: 10.1088/1361-6420/ab2fec.  Google Scholar [10] H. Isozaki, On the existence of solutions of time-dependent Hartree-Fock equations, Publ. Res. Inst. Math. Sci., 19 (1983), 107-115.  doi: 10.2977/prims/1195182978.  Google Scholar [11] R. Kress, Linear Integral Equations, vol. 82 of Applied Mathematical Sciences, 2nd edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0559-3.  Google Scholar [12] J. C. Lemm and J. Uhlig, Hartree-fock approximation for inverse many-body problems, Physical Review Letters, 84 (2000), 4517-4520.  doi: 10.1103/PhysRevLett.84.4517.  Google Scholar [13] C. Lubich, From quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. doi: 10.4171/067.  Google Scholar [14] K. Mochizuki, On small data scattering with cubic convolution nonlinearity, J. Math. Soc. Japan, 41 (1989), 143-160.  doi: 10.2969/jmsj/04110143.  Google Scholar [15] P. M. Morse and H. Feshbach, Methods of Theoretical Physics, 2 volumes, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953.  Google Scholar [16] R. G. Novikov, On inverse scattering for the $N$-body Schrödinger equation, J. Funct. Anal., 159 (1998), 492-536.  doi: 10.1006/jfan.1998.3324.  Google Scholar [17] A. G. Ramm and A. I. Katsevich, The Radon Transform and Local Tomography, CRC Press, Boca Raton, FL, 1996.   Google Scholar [18] W. Strauss, Nonlinear scattering theory, in Scattering Theory in Mathematical Physics (eds. L. J. A. and M. J.-P.), vol. 9 of Nato Advanced Study Institutes Series C, D. Reidel, 1974, 53–78. doi: 10.1007/978-94-010-2147-0_3.  Google Scholar [19] T. Takiguchi, An inverse problem for free channel scattering, in Inverse Problems and Related Topics (Kobe, 1998), vol. 419 of Chapman & Hall/CRC Res. Notes Math., Chapman & Hall/CRC, Boca Raton, FL, 2000,165–179.  Google Scholar [20] G. Uhlmann and A. Vasy, Low-energy inverse problems in three-body scattering, Inverse Problems, 18 (2002), 719-736.  doi: 10.1088/0266-5611/18/3/313.  Google Scholar [21] G. Uhlmann and A. Vasy, Inverse problems in three-body scattering, in Inverse Problems: Theory and Applications (Cortona/Pisa, 2002), vol. 333 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2003,209–215. doi: 10.1090/conm/333/05965.  Google Scholar [22] G. Uhlmann and A. Vasy, Inverse problems in $N$-body scattering, in Inverse Problems and Spectral Theory, vol. 348 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2004,135–154. doi: 10.1090/conm/348/06319.  Google Scholar [23] G. D. Valencia and R. Weder, High-velocity estimates and inverse scattering for quantum $N$-body systems with Stark effect, J. Math. Phys., 53 (2012), 102105, 30pp. doi: 10.1063/1.4757590.  Google Scholar [24] A. Vasy, Structure of the resolvent for three-body potentials, Duke Math. J., 90 (1997), 379-434.  doi: 10.1215/S0012-7094-97-09010-4.  Google Scholar [25] T. Wada, Scattering theory for time-dependent Hartree-Fock type equation, Osaka J. Math., 36 (1999), 905-918.   Google Scholar [26] X. P. Wang, On the uniqueness of inverse scattering for $N$-body systems, Inverse Problems, 10 (1994), 765-784.  doi: 10.1088/0266-5611/10/3/017.  Google Scholar [27] X. P. Wang, High energy asymptotics for n-body scattering matrices with arbitrary channels, Annales de l'IHP Physique Théorique, 65 (1996), 81–108.  Google Scholar [28] M. Watanabe, Inverse scattering for the nonlinear Schrödinger equation with cubic convolution nonlinearity, Tokyo J. Math., 24 (2001), 59-67.  doi: 10.3836/tjm/1255958311.  Google Scholar [29] M. Watanabe, Inverse scattering problem for time dependent Hartree-Fock equations in the three-body case, J. Math. Phys., 48 (2007), 053510, 9pp. doi: 10.1063/1.2732171.  Google Scholar [30] M. Watanabe, A remark on inverse scattering for time dependent hartree equations, in Journal of Physics: Conference Series, 73 (2007), 012025. doi: 10.1088/1742-6596/73/1/012025.  Google Scholar [31] M. Watanabe, Time-dependent method for non-linear Schrödinger equations in inverse scattering problems, J. Math. Anal. Appl., 459 (2018), 932-944.  doi: 10.1016/j.jmaa.2017.11.012.  Google Scholar [32] M. Watanabe, Time-dependent methods in inverse scattering problems for the Hartree-Fock equation, J. Math. Phys., 60 (2019), 091504, 19pp. doi: 10.1063/1.5090924.  Google Scholar [33] R. Weder, Multidimensional inverse scattering in an electric field, J. Funct. Anal., 139 (1996), 441-465.  doi: 10.1006/jfan.1996.0092.  Google Scholar [34] R. Weder, Inverse scattering for $N$-body systems with time-dependent potentials, in Inverse Problems of Wave Propagation and Diffraction (Aix-les-Bains, 1996), vol. 486 of Lecture Notes in Phys., Springer, Berlin, 1997, 27–46. doi: 10.1007/BFb0105758.  Google Scholar [35] R. Weder, Inverse scattering for the nonlinear Schrödinger equation, Comm. Partial Differential Equations, 22 (1997), 2089-2103.  doi: 10.1080/03605309708821332.  Google Scholar [36] R. Weder, Multidimensional inverse problems in perturbed stratified media, J. Differential Equations, 152 (1999), 191-239.  doi: 10.1006/jdeq.1998.3509.  Google Scholar [37] R. Weder, The $W_{k, p}$-continuity of the Schrödinger wave operators on the line, Comm. Math. Phys., 208 (1999), 507-520.  doi: 10.1007/s002200050767.  Google Scholar [38] R. Weder, Inverse scattering on the line for the nonlinear Klein-Gordon equation with a potential, J. Math. Anal. Appl., 252 (2000), 102-123.  doi: 10.1006/jmaa.2000.6954.  Google Scholar [39] R. Weder, $L^p$-$L^{\dot p}$ estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential, J. Funct. Anal., 170 (2000), 37-68.  doi: 10.1006/jfan.1999.3507.  Google Scholar [40] R. Weder, Uniqueness of inverse scattering for the nonlinear Schrödinger equation and reconstruction of the potential and the nonlinearity, in Mathematical and Numerical Aspects of Wave Propagation (Santiago de Compostela, 2000), SIAM, Philadelphia, PA, 2000,631–634.  Google Scholar [41] R. Weder, Inverse scattering for the non-linear Schrödinger equation: reconstruction of the potential and the non-linearity, Math. Methods Appl. Sci., 24 (2001), 245-254.  doi: 10.1002/mma.216.  Google Scholar [42] R. Weder, Inverse scattering for the nonlinear Schrödinger equation. II. Reconstruction of the potential and the nonlinearity in the multidimensional case, Proc. Amer. Math. Soc., 129 (2001), 3637-3645.  doi: 10.1090/S0002-9939-01-06016-6.  Google Scholar [43] R. Weder, Multidimensional inverse scattering for the nonlinear Klein-Gordon equation with a potential, J. Differential Equations, 184 (2002), 62-77.  doi: 10.1006/jdeq.2001.4133.  Google Scholar [44] R. Weder, The $L^p$-$L^{p'}$ estimate for the Schrödinger equation on the half-line, J. Math. Anal. Appl., 281 (2003), 233-243.  doi: 10.1016/S0022-247X(03)00093-3.  Google Scholar [45] K. 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), 311-346.   Google Scholar
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