Papers
Topics
Authors
Recent
Assistant
AI Research Assistant
Well-researched responses based on relevant abstracts and paper content.
Custom Instructions Pro
Preferences or requirements that you'd like Emergent Mind to consider when generating responses.
Gemini 2.5 Flash
Gemini 2.5 Flash 134 tok/s
Gemini 2.5 Pro 41 tok/s Pro
GPT-5 Medium 30 tok/s Pro
GPT-5 High 26 tok/s Pro
GPT-4o 64 tok/s Pro
Kimi K2 185 tok/s Pro
GPT OSS 120B 442 tok/s Pro
Claude Sonnet 4.5 37 tok/s Pro
2000 character limit reached

Feynman-Kac formulas for semigroups generated by multi-polaron Hamiltonians in magnetic fields and on general domains (2403.12147v1)

Published 18 Mar 2024 in math-ph and math.MP

Abstract: We prove Feynman-Kac formulas for the semigroups generated by selfadjoint operators in a class containing Fr\"ohlich Hamiltonians known from solid state physics. The latter model multi-polarons, i.e., a fixed number of quantum mechanical electrons moving in a polarizable crystal and interacting with the quantized phonon field generated by the crystal's vibrational modes. Both the electrons and phonons can be confined to suitable open subsets of Euclidean space. We also include possibly very singular magnetic vector potentials and electrostatic potentials. Our Feynman-Kac formulas comprise Fock space operator-valued multiplicative functionals and can be applied to every vector in the underlying Hilbert space. In comparison to the renormalized Nelson model, for which analogous Feynman-Kac formulas are known, the analysis of the creation and annihilation terms in the multiplicative functionals requires novel ideas to overcome difficulties caused by the phonon dispersion relation being constant. Getting these terms under control and generalizing other construction steps so as to cover confined systems are the main achievements of this article.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (54)
  1. I. Anapolitanos and M. Griesemer. Multipolarons in a Constant Magnetic Field. Ann. Henri Poincaré, 15(6):1037–1059, 2014, arXiv:1204.5660. doi:10.1007/s00023-013-0266-4.
  2. I. Anapolitanos and B. Landon. The Ground State Energy of the Multi-Polaron in the Strong Coupling Limit. Lett. Math. Phys., 103(12):1347–1366, 2013, arXiv:1212.3571. doi:10.1007/s11005-013-0648-z.
  3. M. Aizenman and B. Simon. Brownian Motion and Harnack Inequality for Schrödinger Operators. Commun. Pure Appl. Math., 35(2):209–273, 1982. doi:10.1002/cpa.3160350206.
  4. Continuity properties of Schrödinger semigroups with magnetic fields. Rev. Math. Phys., 12(2):181–225, 2000, arXiv:math-ph/9808004. doi:10.1142/S0129055X00000083.
  5. G. A. Bley. Estimates on Functional Integrals of Non-Relativistic Quantum Field Theory, with Applications to the Nelson and Polaron Models. PhD thesis, University of Virginia, 2016. doi:10.18130/V3QP13.
  6. M. Brooks and D. Mitrouskas. Asymptotic series for low-energy excitations of the Fröhlich Polaron at strong coupling. Preprint, 2023, arXiv:2306.16373.
  7. Effective mass of the Fröhlich Polaron and the Landau-Pekar-Spohn conjecture. Preprint, 2023, arXiv:2307.13058.
  8. V. Betz and S. Polzer. A Functional Central Limit Theorem for Polaron Path Measures. Commun. Pure Appl. Math., 75(11):2345–2392, 2022, arXiv:2106.06447. doi:10.1002/cpa.22080.
  9. V. Betz and S. Polzer. Effective Mass of the Polaron: A Lower Bound. Commun. Math. Phys., 399(1):173–188, 2023, arXiv:2201.06445. doi:10.1007/s00220-022-04553-0.
  10. V. Betz and H. Spohn. A central limit theorem for Gibbs measures relative to Brownian motion. Probab. Theory Relat. Fields, 131(3):459–478, 2005, arXiv:math/0308193. doi:10.1007/s00440-004-0381-8.
  11. Estimates on Functional Integrals of Quantum Mechanics and Non-relativistic Quantum Field Theory. Commun. Math. Phys., 350(1):79–103, 2017, arXiv:1512.00356. doi:10.1007/s00220-017-2834-9.
  12. G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions, volume 152 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2nd edition, 2014. doi:10.1017/CBO9781107295513.
  13. W. Dybalski and H. Spohn. Effective Mass of the Polaron – Revisited. Ann. Henri Poincaré, 21(5):1573–1594, 2020, arXiv:1908.03432. doi:10.1007/s00023-020-00892-7.
  14. Asymptotics for the Polaron. Commun. Pure Appl. Math., 36(4):505–528, 1983. doi:10.1002/cpa.3160360408.
  15. A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, volume 38 of Stochastic Modelling and Applied Probability. Springer, Berlin, 2nd edition, 2010. doi:10.1007/978-3-642-03311-7.
  16. R. P. Feynman. Slow Electrons in a Polar Crystal. Phys. Rev., 97:660–665, 1955. doi:10.1103/PhysRev.97.660.
  17. Stability and absence of binding for multi-polaron systems. Publ. Math. Inst. Hautes Études Sci., 113:39–67, 2011, arXiv:1004.4892. doi:10.1007/s10240-011-0031-5.
  18. H. Föllmer and P. Protter. On Itô’s formula for multidimensional Brownian motion. Probab. Theory Relat. Fields, 116(1):1–20, 2000. doi:10.1007/PL00008719.
  19. H. Fröhlich. Electrons in Lattice Fields. Adv. Phys., 3(11):325–361, 1954. doi:10.1080/00018735400101213.
  20. W. Faris and B. Simon. Degenerate and non-degenerate ground states for Schrödinger operators. Duke Math. J., 42(3):559–567, 1975. doi:10.1215/S0012-7094-75-04251-9.
  21. R. L. Frank and R. Seiringer. Quantum Corrections to the Pekar Asymptotics of a Strongly Coupled Polaron. Commun. Pure Appl. Math., 74(3):544–588, 2021, arXiv:1902.02489. doi:10.1002/cpa.21944.
  22. R. Ghanta. Ground state of the polaron hydrogenic atom in a strong magnetic field. J. Math. Phys., 62(3):031901, 2021, arXiv:1811.12325. doi:10.1063/5.0012192.
  23. Stochastic differential equations for models of non-relativistic matter interacting with quantized radiation fields. Probab. Theory Relat. Fields, 167(3-4):817–915, 2017, arXiv:1402.2242. doi:10.1007/s00440-016-0694-4.
  24. M. Griesemer and D. Wellig. The strong-coupling polaron in static electric and magnetic fields. J. Phys. A, 46(42):425202, 2013, arXiv:1304.2056. doi:10.1088/1751-8113/46/42/425202.
  25. M. Griesemer and A. Wünsch. On the domain of the Fröhlich Hamiltonian. J. Math. Phys., 57(2):021902, 2016, arXiv:1508.02533. doi:10.1063/1.4941561.
  26. F. Hiroshima and J. Lőrinczi. Feynman-Kac-Type Theorems and Gibbs Measures on Path Space. Volume 2: Applications in Rigorous Quantum Field Theory, volume 34/2 of De Gruyter Studies in Mathematics. De Gruyter, Berlin, 2nd edition, 2020. doi:10.1515/9783110403541.
  27. F. Hiroshima and O. Matte. Ground states and their associated path measures in the renormalized Nelson model. Rev. Math. Phys., 34(2):2250002, 2022, arXiv:1903.12024. doi:10.1142/S0129055X22500027.
  28. B. Hinrichs and O. Matte. Feynman–Kac Formula and Asymptotic Behavior of the Minimal Energy for the Relativistic Nelson Model in Two Spatial Dimensions. Preprint, to appear in Ann. Henri Poincaré, 2023, arXiv:2211.14046, doi:10.1007/s00023-023-01369-z.
  29. B. Hinrichs and O. Matte. Feynman–Kac formula for fiber Hamiltonians in the relativistic Nelson model in two spatial dimensions. Preprint, 2023, arXiv:2309.09005.
  30. U. G. Haussmann and É. Pardoux. Time Reversal of Diffusions. Ann. Probab., 14(4):1188–1205, 1986. doi:10.1214/aop/1176992362.
  31. D. Hundertmark. Zur Theorie der magnetischen Schrödingerhalbgruppe. PhD thesis, Ruhr University Bochum, 1996.
  32. V. Liskevich and A. Manavi. Dominated Semigroups with Singular Complex Potentials. J. Funct. Anal., 151(2):281–305, 1997. doi:10.1006/jfan.1997.3150.
  33. H. Löwen. Spectral properties of an optical polaron in a magnetic field. J. Math. Phys., 29(6):1498–1504, 1988. doi:10.1063/1.527893.
  34. J. Lampart and J. Schmidt. On Nelson-Type Hamiltonians and Abstract Boundary Conditions. Commun. Math. Phys., 367(2):629–663, 2019, arXiv:1803.00872. doi:10.1007/s00220-019-03294-x.
  35. E. H. Lieb and K. Yamazaki. Ground-State Energy and Effective Mass of the Polaron. Phys. Rev., 111:728–733, 1958. doi:10.1103/PhysRev.111.728.
  36. O. Matte. Continuity properties of the semi-group and its integral kernel in non-relativistic QED. Rev. Math. Phys., 28(05):1650011, 2016, arXiv:1512.04494. doi:10.1142/S0129055X16500112.
  37. O. Matte. Pauli-Fierz Type Operators with Singular Electromagnetic Potentials on General Domains. Math. Phys. Anal. Geom., 20(2):18, 2017, arXiv:1703.00404. doi:10.1007/s11040-017-9249-x.
  38. O. Matte. Feynman-Kac Formulas for Dirichlet-Pauli-Fierz Operators with Singular Coefficients. Integr. Equ. Oper. Theory, 93(6):62, 2021, arXiv:1906.07616. doi:10.1007/s00020-021-02677-x.
  39. O. Matte and J. S. Møller. Feynman-Kac Formulas for the Ultra-Violet Renormalized Nelson Model. Astérisque, 404, 2018, arXiv:1701.02600. doi:10.24033/ast.1054.
  40. Absence of Absolutely Continuous Spectra for Multidimensional Schrödinger Operators with High Barriers. Bull. London Math. Soc., 27(2):162–168, 1995. doi:10.1112/blms/27.2.162.
  41. C. Mukherjee and S. R. S. Varadhan. Identification of the Polaron measure in strong coupling and the Pekar variational formula. Ann. Probab., 48(5):2119–2144, 2020, arXiv:1812.06927. doi:10.1214/19-AOP1392.
  42. C. Mukherjee and S. R. S. Varadhan. Identification of the Polaron Measure I: Fixed Coupling Regime and the Central Limit Theorem for Large Times. Commun. Pure Appl. Math., 73(2):350–383, 2020, arXiv:1802.05696. doi:10.1002/cpa.21858.
  43. E. Nelson. Interaction of Nonrelativistic Particles with a Quantized Scalar Field. J. Math. Phys., 5(9):1190–1197, 1964. doi:10.1063/1.1704225.
  44. É. Pardoux. Grossissement d’une filtration et retournement du temps d’une diffusion. In Séminaire de Probabilités, XX, 1984/85, volume 1204 of Lecture Notes in Mathematics, pages 48–55. Springer, 1986. doi:10.1007/BFb0075711.
  45. K. R. Parthasarathy. An Introduction to Quantum Stochastic Calculus, volume 85 of Monographs in Mathematics. Birkhäuser, Basel, 1992. doi:10.1007/978-3-0348-0566-7.
  46. S. Polzer. Renewal approach for the energy–momentum relation of the Fröhlich polaron. Lett. Math. Phys., 113(4):90, 2023, arXiv:2206.14425. doi:10.1007/s11005-023-01711-w.
  47. A. Posilicano. On the Self-Adjointness of H+A*+A𝐻superscript𝐴𝐴H+A^{*}+Aitalic_H + italic_A start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT + italic_A. Math. Phys. Anal. Geom., 23(4):37, 2020, arXiv:2003.05412. doi:10.1007/s11040-020-09359-x.
  48. M. Reed and B. Simon. Functional Analysis, volume 1 of Methods of Modern Mathematical Physics. Academic Press, San Diego, revised and enlarged edition, 1980.
  49. B. Simon. A canonical decomposition for quadratic forms with applications to monotone convergence theorems. J. Funct. Anal., 28(3):377–385, 1978. doi:10.1016/0022-1236(78)90094-0.
  50. B. Simon. Classical boundary conditions as a technical tool in modern mathematical physics. Adv. Math., 30(3):268–281, 1978. doi:10.1016/0001-8708(78)90040-3.
  51. B. Simon. Maximal and minimal Schrödinger forms. J. Operator Theory, 1(1):37–47, 1979.
  52. B. Simon. Schrödinger semigroups. Bull. Amer. Math. Soc., 7(3):447–526, 1982. doi:10.1090/S0273-0979-1982-15041-8.
  53. H. Spohn. Effective mass of the polaron: A functional integral approach. Ann. Phys., 175(2):278–318, 1987. doi:10.1016/0003-4916(87)90211-9.
  54. Elliptische Differentialgleichungen zweiter Ordnung. Springer, Dordrecht, 2009. doi:10.1007/978-3-540-45721-3.
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
Dice Question Streamline Icon: https://streamlinehq.com

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now
Lightbulb Streamline Icon: https://streamlinehq.com

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now
List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets

This paper has been mentioned in 1 tweet and received 0 likes.

Upgrade to Pro to view all of the tweets about this paper:

Start a free 7-day Pro trial