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Research Article

Standing waves and global well-posedness for the 2d Hartree equation with a point interaction

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Pages 242-278 | Received 12 Apr 2022, Accepted 18 Feb 2024, Published online: 27 Apr 2024

References

  • Pickl, P. (2011). A simple derivation of mean field limits for quantum systems. Lett. Math. Phys. 97:151–164. DOI: 10.1007/s11005-011-0470-4.
  • Benedikter, N., Porta, M., Schlein, B. (2016). Effective Evolution Equations from Quantum Dynamics. Vol. 7 of Springer Briefs in Mathematical Physics. Cham: Springer.
  • Leopold, N., Pickl, P. (2020). Derivation of the Maxwell-Schrödinger equations from the Pauli-Fierz Hamiltonian. SIAM J. Math. Anal. 52:4900–4936. DOI: 10.1137/19M1307639.
  • Coclite, G. M., Georgiev, V. (2004). Solitary waves for Maxwell-Schrödinger equations. Electron. J. Differ. Equ. 94:1–31.
  • Nakamura, M., Wada, T. (2007). Global existence and uniqueness of solutions to the Maxwell-Schrödinger equations. Commun. Math. Phys. 276:315–339. DOI: 10.1007/s00220-007-0337-9.
  • Ginibre, J., Velo, G. (2008). Uniqueness at infinity in time for the Maxwell-Schrödinger system with arbitrarily large asymptotic data. Port. Math. 65:509–534. DOI: 10.4171/pm/1824.
  • Colin, M., Watanabe, T. (2019). A refined stability result for standing waves of the Schrödinger-Maxwell system. Nonlinearity 32:3695–3714. DOI: 10.1088/1361-6544/ab248c.
  • Antonelli, P., Marcati, P., Scandone, R. (2022). Global well-posedness for the non-linear Maxwell-Schrödinger system. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5:1293–1324. DOI: 10.2422/2036-2145.202010_033.
  • Adami, R., Sacchetti, A. (2005). The transition from diffusion to blow-up for a nonlinear Schrödinger equation in dimension 1. J. Phys. A 38:8379–8392. DOI: 10.1088/0305-4470/38/39/006.
  • Della Casa, F. F. G., Sacchetti, A. (2006). Stationary states for non linear one-dimensional Schrödinger equations with singular potential. Phys. D 219:60–68. DOI: 10.1016/j.physd.2006.05.014.
  • Fukuizumi, R., Jeanjean, L. (2008). Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential. Discrete Contin. Dyn. Syst. 21:121–136. DOI: 10.3934/dcds.2008.21.121.
  • Fukuizumi, R., Ohta, M., Ozawa, T. (2008). Nonlinear Schrödinger equation with a point defect. Ann. Inst. H.Poincaré Anal. Non Linéaire 25:837–845. DOI: 10.1016/j.anihpc.2007.03.004.
  • Le Coz, S., Fukuizumi, R., Fibich, G., Ksherim, B., Yonatan, S. (2008). Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential. Phys. D 237:1103–1128. DOI: 10.1016/j.physd.2007.12.004.
  • Adami, R., Noja, D. (2009). Existence of dynamics for a 1D NLS equation perturbed with a generalized point defect. J. Phys. A 42:495302, 19. DOI: 10.1088/1751-8113/42/49/495302.
  • Adami, R., Noja, D., Visciglia, N. (2013). Constrained energy minimization and ground states for NLS with point defects. Discrete Contin. Dyn. Syst. Ser. B 18:1155–1188. DOI: 10.3934/dcdsb.2013.18.1155.
  • Adami, R., Noja, D. (2014). Exactly solvable models and bifurcations: the case of the cubic NLS with a δ or a δ′ interaction in dimension one. Math. Model. Nat. Phenom. 9:1–16.
  • Angulo Pava, J., Ferreira, L. C. F. (2014). On the Schrödinger equation with singular potentials. Differ. Integral Equ. 27:767–800.
  • Banica, V., Visciglia, N. (2016). Scattering for NLS with a delta potential. J. Differ. Equ. 260:4410–4439. DOI: 10.1016/j.jde.2015.11.016.
  • Ikeda, M., Inui, T. (2017). Global dynamics below the standing waves for the focusing semilinear Schrödinger equation with a repulsive Dirac delta potential. Anal. PDE 10:481–512. DOI: 10.2140/apde.2017.10.481.
  • Ianni, I., Le Coz, S., Royer, J. (2017). On the Cauchy problem and the black solitons of a singularly perturbed Gross-Pitaevskii equation. SIAM J. Math. Anal. 49:1060–1099. DOI: 10.1137/15M1029606.
  • Cuccagna, S., Maeda, M. (2019). On stability of small solitons of the 1-D NLS with a trapping delta potential. SIAM J. Math. Anal. 51:4311–4331. DOI: 10.1137/19M1258402.
  • Angulo Pava, J., Hernández Melo, C. A. (2019). On stability properties of the cubic-quintic Schrödinger equation with δ -point interaction. Commun. Pure Appl. Anal. 18:2093–2116.
  • Masaki, S., Murphy, J., Segata, J.-I. (2019). Modified scattering for the one-dimensional cubic NLS with a repulsive delta potential. Int. Math. Res. Not. IMRN 2019:7577–7603. DOI: 10.1093/imrn/rny011.
  • Masaki, S., Murphy, J., Segata, J.-I. (2020). Stability of small solitary waves for the one-dimensional NLS with an attractive delta potential. Anal. PDE 13:1099–1128. DOI: 10.2140/apde.2020.13.1099.
  • Masaki, S., Murphy, J., Segata, J.-I. (2021). A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete Contin. Dyn. Syst. Ser. S 14:1693–1716.
  • Georgiev, V., Michelangeli, A., Scandone, R. (2018). On fractional powers of singular perturbations of the Laplacian. J. Funct. Anal. 275:1551–1602. DOI: 10.1016/j.jfa.2018.03.007.
  • Michelangeli, A., Olgiati, A., Scandone, R. (2018). Singular Hartree equation in fractional perturbed Sobolev spaces. J. Nonlinear Math. Phys. 25:558–588. DOI: 10.1080/14029251.2018.1503423.
  • Cacciapuoti, C., Finco, D., Noja, D. (2021). Well posedness of the nonlinear Schrödinger equation with isolated singularities. J. Differ. Equ. 305:288–318. DOI: 10.1016/j.jde.2021.10.017.
  • Adami, R., Boni, F., Carlone, R., Tentarelli, L. (2022). Ground states for the planar NLSE with a point defect as minimizers of the constrained energy. Calc. Var. 61:195. DOI: 10.1007/s00526-022-02310-8.
  • Adami, R., Boni, F., Carlone, R., Tentarelli, L. (2022). Existence, structure, and robustness of ground states of a NLSE in 3D with a point defect. J. Math. Phys. 63:071501.
  • Fukaya, N., Georgiev, V., Ikeda, M. (2022). On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction. J. Differ. Equ. 321:258–295. DOI: 10.1016/j.jde.2022.03.008.
  • Boni, F., Gallone, M. (2023). Two dimensional NLS ground states with attractive Coulomb potential and point interaction. arXiv:2304.13629.
  • Boni, F., Gallone, M. (2020). Erratum: Two-dimensional Schrödinger operators with point interactions: threshold expansions, zero modes and Lp -boundedness of wave operators. Rev. Math. Phys. 32:2092001, 5.
  • Cornean, H. D., Michelangeli, A., Yajima, K. (2019). Two-dimensional Schrödinger operators with point interactions: threshold expansions, zero modes and Lp -boundedness of wave operators. Rev. Math. Phys. 31:1950012, 32.
  • Yajima, K. (2021). Lp -boundedness of wave operators for 2D Schrödinger operators with point interactions. Ann. Henri Poincaré 22:2065–2101.
  • Yajima, K. (2005). Solvable Models in Quantum Mechanics. 2nd ed. Providence, RI: AMS Chelsea Publishing. With an appendix by Pavel Exner.
  • Albeverio, S., Gesztesy, F., Hø egh-Krohn, R., Holden, H. (1987). Point interactions in two dimensions: basic properties, approximations and applications to solid state physics. J. Reine Angew. Math. 380:87–107.
  • Michelangeli, A., Scandone, R. (2019). Point-like perturbed fractional Laplacians through shrinking potentials of finite range. Complex Anal. Oper. Theory 13:3717–3752. DOI: 10.1007/s11785-019-00927-w.
  • Gallone, M., Michelangeli, A., Ottolini, A. (2020). Kreĭn-Višik-Birman self-adjoint extension theory revisited. In: Michelangeli, A., ed. Mathematical Challenges of Zero Range Physics. INdAM-Springer Series, Vol. 42. Cham: Springer, pp. 239–304.
  • (2023). Self-Adjoint Extension Schemes and Modern Applications to Quantum Hamiltonians. Cham: Springer. Foreword by Sergio Albeverio.
  • Kesavan, S. (2006). Symmetrization & Applications. Vol. 3 of Series in Analysis. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd.
  • Cazenave, T. (2003). Semilinear Schrödinger equations. Vol. 10 of Courant Lecture Notes in Mathematics. New York: New York University Courant Institute of Mathematical Sciences.
  • Michelangeli, A. (2015). Global well-posedness of the magnetic Hartree equation with non-Strichartz external fields. Nonlinearity 28:2743–2765. DOI: 10.1088/0951-7715/28/8/2743.
  • Antonelli, P., Michelangeli, A., Scandone, R. (2018). Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials. Z. Angew. Math. Phys. 69:46. DOI: 10.1007/s00033-018-0938-5.
  • Okazawa, N., Suzuki, T., Yokota, T. (2012). Energy methods for abstract nonlinear Schrödinger equations. Evol. Equ. Control Theory 1:337–354. DOI: 10.3934/eect.2012.1.337.
  • Abramowitz, M., Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Vol. 55 of National Bureau of Standards Applied Mathematics Series, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.
  • Strauss, W. A. (1977). Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55:149–162. DOI: 10.1007/BF01626517.
  • Bergh, J., Löfström, J. (1976). Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Berlin-New York: Springer-Verlag.
  • D’Ancona, P., Pierfelice, V., Teta, A. (2006). Dispersive estimate for the Schrödinger equation with point interactions. Math. Methods Appl. Sci. 29:309–323. DOI: 10.1002/mma.682.
  • Duchêne, V., Marzuola, J. L., Weinstein, M. I. (2011). Wave operator bounds for one-dimensional Schrödinger operators with singular potentials and applications. J. Math. Phys. 52:013505, 17.
  • Iandoli, F., Scandone, R. (2017). Dispersive estimates for Schrödinger operators with point interactions in R3 . In: Michelangeli, A., Dell’Antonio, G., eds. Advances in Quantum Mechanics: Contemporary Trends and Open Problems. Springer INdAM Series, Vol. 18. Cham: Springer, pp. 187–199.
  • Dell’Antonio, G., Michelangeli, A., Scandone, R., Yajima, K. (2018). Lp -Boundedness of Wave Operators for the Three-Dimensional Multi-Centre Point Interaction. Ann. Henri Poincaré 19:283–322.
  • Christ, M., Kiselev, A. (2001). Maximal functions associated to filtrations. J. Funct. Anal. 179:409–425. DOI: 10.1006/jfan.2000.3687.
  • Luttinger, J. M., Friedberg, R. (1976). A new rearrangement inequality for multiple integrals. Arch. Rational Mech. Anal. 61:45–64. DOI: 10.1007/BF00251862.
  • Brascamp, H. J., Lieb, E. H., Luttinger, J. M. (1974). A general rearrangement inequality for multiple integrals. J. Funct. Anal. 17:227–237. DOI: 10.1016/0022-1236(74)90013-5.
  • Burchard, A., Ferone, A. (2015). On the extremals of the Pólya-Szegő inequality. Indiana Univ. Math. J. 64:1447–1463. DOI: 10.1512/iumj.2015.64.5652.
  • Brothers, J. E., Ziemer, W. P. (1988). Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384:153–179.
  • Aronszajn, N., Smith, K. T. (1961). Theory of Bessel potentials. I. Ann. Inst. Fourier (Grenoble) 11:385–475. DOI: 10.5802/aif.116.

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