×

A note on the Fröhlich dynamics in the strong coupling limit. (English) Zbl 1462.81083

Summary: We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in [M. Griesemer, Rev. Math. Phys. 29, No. 10, Article ID 1750030, 21 p. (2017; Zbl 1377.81264)]. In the latter it was shown that the Fröhlich time evolution applied to the initial state \(\varphi_0\otimes \xi_\alpha\), where \(\varphi_0\) is the electron ground state of the Pekar energy functional and \(\xi_\alpha\) the associated coherent state of the phonons, can be approximated by a global phase for times small compared to \(\alpha^2\). In the present note we prove that a similar approximation holds for \(t=O(\alpha^2)\) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to \(\alpha^{-2}\) and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order \(\alpha^2\), while the phonon fluctuations around the coherent state \(\xi_\alpha\) can be described by a time-dependent Bogoliubov transformation.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q15 Perturbation theories for operators and differential equations in quantum theory
82D20 Statistical mechanics of solids

Citations:

Zbl 1377.81264
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alexandrov, AS; Devreese, JT, Advances in Polaron Physics (2010), New York: Springer, New York · doi:10.1007/978-3-642-01896-1
[2] Boßmann, L., Petrat, S., Pickl, P., Soffer, A.: Beyond Bogoliubov Dynamics. Preprint arXiv:1912.11004 (2019)
[3] Donsker, MD; Varadhan, SRS, Asymptotics for the polaron, Commun. Pure Appl. Math., 36, 505-528 (1983) · Zbl 0538.60081 · doi:10.1002/cpa.3160360408
[4] Falconi, M.: Self-adjointness criterion for operators in Fock spaces. Math. Phys. Anal. Geom. 18, Art. 2 (2015) · Zbl 1337.47033
[5] Faris, WG; Lavine, RB, Commutators and self-adjointness of Hamiltonian operators, Commun. Math. Phys., 35, 39-48 (1974) · Zbl 0287.47004 · doi:10.1007/BF01646453
[6] Feliciangeli, D.; Rademacher, S.; Seiringer, R., Persistence of the spectral gap for the Landau-Pekar equations, Lett. Math. Phys., 111, 19 (2021) · Zbl 1469.35183 · doi:10.1007/s11005-020-01350-5
[7] Frank, RL; Schlein, B., Dynamics of a strongly coupled polaron, Lett. Math. Phys., 104, 911-929 (2014) · Zbl 1297.35202 · doi:10.1007/s11005-014-0700-7
[8] Frank, RL; Gang, Z., Derivation of an effective evolution equation for a strongly coupled polaron, Anal. PDE, 10, 379-422 (2017) · Zbl 1365.35130 · doi:10.2140/apde.2017.10.379
[9] Frank, RL; Gang, Z., A non-linear adiabatic theorem for the Landau-Pekar equations, Oberwolfach Rep. (2017) · Zbl 1445.35128 · doi:10.1016/j.jfa.2020.108631
[10] Frank, RL; Gang, Z., A non-linear adiabatic theorem for the one-dimensional Landau-Pekar equations, J. Funct. Anal., 279, 7 (2020) · Zbl 1445.35128 · doi:10.1016/j.jfa.2020.108631
[11] Frank, RL; Seiringer, R., Quantum corrections to the Pekar asymptotics of a strongly coupled polaron, Commun. Pure Appl. Math. (2020) · Zbl 1467.82092 · doi:10.1002/cpa.21944
[12] Fröhlich, H., Theory of electrical breakdown in ionic crytals, Proc. R. Soc. Lond. A, 160, 230-241 (1937) · doi:10.1098/rspa.1937.0106
[13] Griesemer, M., On the dynamics of polarons in the strong-coupling limit, Rev. Math. Phys., 29, 10, 1750030 (2017) · Zbl 1377.81264 · doi:10.1142/S0129055X17500301
[14] Griesemer, M.; Wünsch, A., Self-adjointness and domain of the Fröhlich Hamiltonian, J. Math. Phys., 57, 2 (2016) · Zbl 1342.82140 · doi:10.1063/1.4941561
[15] Jeblick, M.; Mitrouskas, D.; Petrat, S.; Pickl, P., Free time evolution of a tracer particle coupled to a fermi gas in the high-density limit, Commun. Math. Phys., 356, 143-187 (2017) · Zbl 1380.82046 · doi:10.1007/s00220-017-2970-2
[16] Jeblick, M., Mitrouskas, D., Pickl, P.: Effective dynamics of two tracer particles coupled to a fermi gas in the high-density limit. In: Macroscopic Limits of Quantum Systems, Springer Proceedings in Mathematics & Statistics (2017) · Zbl 1414.82031
[17] Lampart, J.; Schmidt, J., On Nelson-type hamiltonians and abstract boundary conditions, Commun. Math. Phys., 367, 629-663 (2019) · Zbl 1414.81285 · doi:10.1007/s00220-019-03294-x
[18] Leopold, N., Mitrouskas, D., Rademacher, S., Schlein, B., Seiringer, R.: Landau-Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron. Preprint. arXiv:2005.02098 (2020)
[19] Leopold, N., Rademacher, S., Schlein, B., Seiringer, R.: The Landau-Pekar equations: adiabatic theorem and accuracy. Preprint. arXiv:1904.12532, Analysis & PDE (in press)
[20] Lieb, EH, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math., 57, 93-105 (1977) · Zbl 0369.35022 · doi:10.1002/sapm197757293
[21] Lieb, E.H., Thomas, L.E.: Exact ground state energy of the strong-coupling polaron. Commun. Math. Phys. 183, 511-519 (1997). Erratum: ibid. 188, 499 (1997) · Zbl 0874.60095
[22] Lieb, EH; Yamazaki, K., Ground-state energy and effective mass of the polaron, Phys. Rev., 111, 728-733 (1958) · Zbl 0100.42504 · doi:10.1103/PhysRev.111.728
[23] Nam, PT; Napiórkowski, M., Bogoliubov correction to the mean-field dynamics of interacting bosons, Adv. Theor. Math. Phys., 21, 683-738 (2017) · Zbl 1382.82032 · doi:10.4310/ATMP.2017.v21.n3.a4
[24] Pekar, SI, Untersuchung über die Elektronentheorie der Kristalle (1954), Akad: Berlin. Verlag, Akad · Zbl 0058.45503
[25] Reed, M.; Simon, B., Methods of Modern Mathematical Physics, Analysis of Operators (1978), New York: Academic Press, New York · Zbl 0401.47001
[26] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Functional Analysis, vol. I. Academic Press, New York (1980) · Zbl 0459.46001
[27] Solovej, J.P.: Many body quantum mechanics. Lecture notes. https://www.mathematik.uni-muenchen.de/ sorensen/Lehre/SoSe2013/MQM2/skript.pdf (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.