We prove an adiabatic theorem for the Landau–Pekar equations. This allows us to derive new results on the accuracy of their use as effective equations for the time evolution generated by the Fröhlich Hamiltonian with large coupling constant α. In particular, we show that the time evolution of Pekar product states with coherent phonon field and the electron being trapped by the phonons is well approximated by the Landau–Pekar equations until times short compared to α2.
Analysis and PDE
N. L. and R. S. gratefully acknowledge financial support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 694227). B. S. acknowledges support from the Swiss National Science Foundation (grant 200020_172623) and from the NCCR SwissMAP. N. L. would like to thank Andreas Deuchert and David Mitrouskas for interesting discussions. B. S. and R. S. would like to thank Rupert Frank for stimulating discussions about the time-evolution of a polaron.
Leopold NK, Rademacher SAE, Schlein B, Seiringer R. The Landau–Pekar equations: Adiabatic theorem and accuracy. Analysis and PDE. 2021;14(7):2079-2100. doi:10.2140/APDE.2021.14.2079
Leopold, N. K., Rademacher, S. A. E., Schlein, B., & Seiringer, R. (2021). The Landau–Pekar equations: Adiabatic theorem and accuracy. Analysis and PDE. Mathematical Sciences. https://doi.org/10.2140/APDE.2021.14.2079
Leopold, Nikolai K, Simone Anna Elvira Rademacher, Benjamin Schlein, and Robert Seiringer. “ The Landau–Pekar Equations: Adiabatic Theorem and Accuracy.” Analysis and PDE. Mathematical Sciences, 2021. https://doi.org/10.2140/APDE.2021.14.2079.
N. K. Leopold, S. A. E. Rademacher, B. Schlein, and R. Seiringer, “ The Landau–Pekar equations: Adiabatic theorem and accuracy,” Analysis and PDE, vol. 14, no. 7. Mathematical Sciences, pp. 2079–2100, 2021.
Leopold NK, Rademacher SAE, Schlein B, Seiringer R. 2021. The Landau–Pekar equations: Adiabatic theorem and accuracy. Analysis and PDE. 14(7), 2079–2100.
Leopold, Nikolai K., et al. “ The Landau–Pekar Equations: Adiabatic Theorem and Accuracy.” Analysis and PDE, vol. 14, no. 7, Mathematical Sciences, 2021, pp. 2079–100, doi:10.2140/APDE.2021.14.2079.