Abstract
We consider the dynamics of the mean-field polaron in the limit of infinite phonon frequency \(\omega \rightarrow \infty \). This is a singular limit formally leading to a Schrödinger–Poisson system that is equivalent to the nonlinear Choquard equation. By establishing estimates between the approximation obtained via the Choquard equation and true solutions of the original system we show that the Choquard equation makes correct predictions about the dynamics of the polaron mean-field model for small values of \(\varepsilon = 1/\omega \).
Similar content being viewed by others
References
Ammari, Z., Falconi, M.: Wigner measures approach to the classical limit of the Nelson model: convergence of dynamics and ground state energy. J. Stat. Phys. 157(2), 330–362 (2014)
Bechouche, P., Nieto, J., Ruiz Arriola, E., Soler, J.: On the time evolution of the mean-field polaron. J. Math. Phys. 41(7), 4293–4312 (2000)
Devreese, J.T., Alexandrov, A.S.: Fröhlich polaron and bipolaron: recent developments. Rep. Prog. Phys. 72(6), 066501 (2009)
Erdős, L., Yau, H.-T.: Derivation of the nonlinear Schrödinger equation from a many body Coulomb system. Adv. Theor. Math. Phys. 5(6), 1169–1205 (2001)
Frank, R.L., Gang, Z.: Derivation of an effective evolution equation for a strongly coupled polaron. arXiv:1505.03059
Frank, R.L., Schlein, B.: Dynamics of a strongly coupled polaron. Lett. Math. Phys. 104(8), 911–929 (2014)
Friedlander, F.G.: Introduction to the Theory of Distributions, 2nd edn. Cambridge University Press, Cambridge (1998). (with additional material by M. Joshi)
Fröhlich, J., Tsai, T.-P., Yau, H.-T.: On the point-particle (Newtonian) limit of the non-linear Hartree equation. Commun. Math. Phys. 225(2), 223–274 (2002)
Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations with nonlocal interaction. Math. Z. 170(2), 109–136 (1980)
Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57(2), 93–105 (1976/1977)
Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195(2), 455–467 (2010)
Miao, C., Xu, G., Zhao, L.: The Cauchy problem of the Hartree equation. J. Partial Differ. Equ. 21(1), 22–44 (2008)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)
Penrose, R.: Quantum computation, entanglement and state reduction. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 356(1743), 1927–1939 (1998)
Schneider, G.: Validity and limitation of the Newell–Whitehead equation. Math. Nachr. 176, 249–263 (1995)
Schneider, G., Sunny, D.A., Zimmermann, D.: The NLS approximation makes wrong predictions for the water wave problem in case of small surface tension and spatially periodic boundary conditions. J. Dyn. Differ. Equ. 27(3), 1077–1099 (2015)
Acknowledgements
This work of the authors is partially supported by the Deutsche Forschungsgemeinschaft DFG through the Graduiertenkolleg GRK 1838 “Spectral Theory and Dynamics of Quantum Systems”.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Griesemer, M., Schmid, J. & Schneider, G. On the dynamics of the mean-field polaron in the high-frequency limit. Lett Math Phys 107, 1809–1821 (2017). https://doi.org/10.1007/s11005-017-0969-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-017-0969-4