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Articles

The Hartree–Fock equations in modulation spaces

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Pages 1088-1117 | Received 04 Jul 2019, Accepted 13 Apr 2020, Published online: 05 May 2020

References

  • Gross, E. P. (1961). Structure of a quantized vortex in boson systems. Nuovo Cim. 20(3):454–477. DOI: 10.1007/BF02731494.
  • Pitaevskii, L. P. (1961). Vortex lines in an imperfect Bose gas. Sov. Phys. JETP. 13(2):451–454.
  • Elgart, A., Schlein, B. (2007). Mean field dynamics of boson stars. Comm. Pure Appl. Math. 60(4):500–545. DOI: 10.1002/cpa.20134.
  • Lenzmann, E. (2007). Well-posedness for semi-relativistic Hartree equations of critical type. Math. Phys. Anal. Geom. 10(1):43–64. DOI: 10.1007/s11040-007-9020-9.
  • Fock, V. (1930). Näherungsmethode zur lösung des quantenmechanischen Mehrkörperproblems. Z Physik. 61(1-2):126–148. DOI: 10.1007/BF01340294.
  • Fröhlich, J., Lenzmann, E. (2007). Dynamical collapse of white dwarfs in Hartree-and Hartree-Fock theory. Commun. Math. Phys. 274(3):737–750. DOI: 10.1007/s00220-007-0290-7.
  • Lipparini, E. (2008). Modern Many-Particle Physics: Atomic Gasses, Nanostructures and Quantum Liquids. 2nd ed. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd.
  • Cabré, X., Sire, Y. (2014). Nonlinear equations for fractional laplacians, I: Regularity, maximum principles, and hamiltonian estimates. Ann. LHenri Poincare (C) Non Linear Anal. 31(1):23–53. DOI: 10.1016/j.anihpc.2013.02.001.
  • Laskin, N. (2002). Fractional Schrödinger equation. Phys. Rev. E. 66(5):056108. DOI: 10.1103/PhysRevE.66.056108.
  • Thangavelu, S. (1993). Lectures on Hermite and Laguerre Expansions, vol. 42. Princeton, NJ: Princeton University Press.
  • Carles, R. (2003). Nonlinear Schrödinger equations with repulsive harmonic potential and applications. SIAM J. Math. Anal. 35(4):823–843. DOI: 10.1137/S0036141002416936.
  • Chadam, J. M., Glassey, R. T. (1975). Global existence of solutions to the Cauchy problem for time-dependent Hartree equations. J. Math. Phys. 16(5):1122–1130. DOI: 10.1063/1.522642.
  • Komech, A. (2015). On the Hartree-Fock dynamics in wave-matrix picture. Dyn. Partial Diff. Eq. 12(2):157–176. DOI: 10.4310/DPDE.2015.v12.n2.a4.
  • Lewin, M., Sabin, J. (2014). The Hartree equation for infinitely many particles, II: Dispersion and scattering in 2D. Anal. PDE . 7(6):1339–1363. DOI: 10.2140/apde.2014.7.1339.
  • Lewin, M., Sabin, J. (2015). The Hartree equation for infinitely many particles I. Well-posedness theory. Commun. Math. Phys. . 334(1):117–170.
  • Bove, A., Da Prato, G., Fano, G. (1974). An existence proof for the Hartree-Fock time-dependent problem with bounded two-body interaction. Communmath. Phys. 37(3):183–191. DOI: 10.1007/BF01646344.
  • Bove, A., Da Prato, G., Fano, G. (1976). On the Hartree-Fock time-dependent problem. Commun. Math. Phys. 49(1):25–33.
  • Chadam, J. M. (1976). The time-dependent Hartree-Fock equations with Coulomb two-body interaction. Communmath. Phys. . 46(2):99–104. DOI: 10.1007/BF01608490.
  • Carles, R., Lucha, W., Moulay, E. (2015). Higher-order Schrödinger and Hartree–Fock equations. J. Math. Phys. . 56(12):122301. DOI: 10.1063/1.4936646.
  • Hainzl, C., Schlein, B. (2009). Stellar collapse in the time dependent Hartree-Fock approximation. Commun. Math. Phys. 287(2):705–717. DOI: 10.1007/s00220-008-0668-1.
  • Baoxiang, W., Lifeng, Z., Boling, G. (2006). Isometric decomposition operators, function spaces Eλp,q and Applications to nonlinear evolution equations. J. Funct. Anal. 233(1):1–39.
  • Bényi, Á., Gröchenig, K., Okoudjou, K. A., Rogers, L. G. (2007). Unimodular Fourier multipliers for modulation spaces. J. Funct. Anal. 246(2):366–384. DOI: 10.1016/j.jfa.2006.12.019.
  • Bényi, Á., Okoudjou, K. A. (2009). Local well-posedness of nonlinear dispersive equations on modulation spaces. Bull. London Math. Soc. 41(3):549–558. DOI: 10.1112/blms/bdp027.
  • Bhimani, D. G. (2016). The Cauchy problem for the Hartree type equation in modulation spaces. Nonlinear Anal. 130:190–201. DOI: 10.1016/j.na.2015.10.002.
  • Bhimani, D. G. (2019). Global well-posedness for fractional Hartree equation on modulation spaces and Fourier algebra. J. Differ. Equ. . 268(1):141–159.
  • Ruzhansky, M., Wang, B., Zhang, H. (2016). Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces. J. Math. Pures Appl. 105(1):31–65. DOI: 10.1016/j.matpur.2015.09.005.
  • Wang, B., Han, L., Huang, C. (2009). Global well-posedness and scattering for the derivative nonlinear Schrödinger equation with small rough data. Ann. IHP Anal. Non Linéaire. 26(6):2253–2281. DOI: 10.1016/j.anihpc.2009.03.004.
  • Wang, B., Hudzik, H. (2007). The global Cauchy problem for the NLS and NLKG with small rough data. J. Differ. Eq. 232(1):36–73. DOI: 10.1016/j.jde.2006.09.004.
  • Wang, B., Huo, Z., Guo, Z., Hao, C. (2011). Harmonic Analysis Method for Nonlinear Evolution Equations, I. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd.
  • Ruzhansky, M., Sugimoto, M., Wang, B. (2012). Modulation Spaces and Nonlinear Evolution Equations. Evolution equations of hyperbolic and Schrödinger type. Basel AG, Basel: Birkhäuser/Springer, p. 267–283.
  • Guo, Z., Wang, Y. (2014). Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear schrödinger and wave equations. JAMA. 124(1):1–38. DOI: 10.1007/s11854-014-0025-6.
  • Kato, K., Naumkin, I. (2019). Estimates on the modulation spaces for the Dirac equation with potential. Rev. Mat. Complut. . 32(2):305–325. DOI: 10.1007/s13163-018-0284-3.
  • Bhimani, D. G. (2019). The nonlinear Schrödinger equations with harmonic potential in modulation spaces. Discr. Cont. Dyn. Syst. A. 39(10):5923–5944.
  • Feichtinger, H. G. (1983). Modulation Spaces on Locally Compact Abelian Groups. In: Krishna, M., Radha, R., Thangavelu, S., eds. Wavelets and Their Applications. Technical Report, University of Vienna. New Delhi, India: Allied Publishers, pp. 99–140.
  • Gröchenig, K. (2013). Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Boston, MA: Birkhäuser Boston, Inc.,/Springer.
  • Toft, J. (2004). Continuity properties for modulation spaces, with applications to pseudo-differential calculus-I. J. Funct. Anal. 207(2):399–429. DOI: 10.1016/j.jfa.2003.10.003.
  • Cunanan, J., Kobayashi, M., Sugimoto, M. (2015). Inclusion relations between Lp-Sobolev and Wiener amalgam spaces. J. Funct. Anal. 268(1):239–254. DOI: 10.1016/j.jfa.2014.10.017.
  • Kobayashi, M., Sugimoto, M. (2011). The inclusion relation between Sobolev and modulation spaces. J. Funct. Anal. 260(11):3189–3208. DOI: 10.1016/j.jfa.2011.02.015.
  • Deng, Q., Ding, Y., Sun, L. (2013). Estimate for generalized unimodular multipliers on modulation spaces. Nonlinear Anal. Theory Methods Appl. 85:78–92. DOI: 10.1016/j.na.2013.02.008.
  • Chen, D., Fan, L. Sun, J. (2012). Asymptotic estimates for unimodular Fourier multipliers on modulation spaces. Discr. Cont. Dyn. Syst. A. 32(2):467–485., DOI: 10.3934/dcds.2012.32.467.
  • Keel, M., Tao, T. (1998). Endpoint Strichartz estimates. Am. J. Math. 120(5):955–980. DOI: 10.1353/ajm.1998.0039.
  • Bhimani, D. G., Balhara, R., and S. Thangavelu, (2019). Hermite multipliers on modulation spaces. In: Delgado, J., Ruzhansky M., eds. Analysis and Partial Differential Equations: Perspectives from Developing Countries. Springer Proceedings in Mathematics & Statistics, vol 275. Cham: Springer.
  • Cordero, E., Nicola, F. (2008). Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation. J. Funct. Anal. 254(2):506–534. DOI: 10.1016/j.jfa.2007.09.015.
  • Carles, R. (2011). Nonlinear Schrödinger equation with time dependent potential. Commun. Math. Sci. 9(4):937–964.

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