Research PaperNo Access

Bogoliubov dynamics and higher-order corrections for the regularized Nelson model

Marco Falconi

Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

E-mail Address: marco.falconi@polimi.it

Corresponding author.

Search for more papers by this author

Nikolai Leopold

Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland

E-mail Address: nikolai.leopold@unibas.ch

Search for more papers by this author

David Mitrouskas

Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria

E-mail Address: david.mitrouskas@ist.ac.at

Search for more papers by this author

, and

Sören Petrat

School of Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany

E-mail Address: s.petrat@jacobs-university.de

Search for more papers by this author

https://doi.org/10.1142/S0129055X2350006XCited by:3 (Source: Crossref)

Abstract

We study the time evolution of the Nelson model in a mean-field limit in which $N$ nonrelativistic bosons weakly couple (with respect to the particle number) to a positive or zero mass quantized scalar field. Our main result is the derivation of the Bogoliubov dynamics and higher-order corrections. More precisely, we prove the convergence of the approximate wave function to the many-body wave function in norm, with a convergence rate proportional to the number of corrections taken into account in the approximation. We prove an analogous result for the unitary propagator. As an application, we derive a simple system of partial differential equations describing the time evolution of the first- and second-order approximations to the one-particle reduced density matrices of the particles and the quantum field, respectively.

Keywords:

AMSC: 35Q40, 35Q55, 81Q05, 81T10, 81V73, 82C10

References

1. A. Abdesselam and D. Hasler , Analyticity of the ground state energy for massless Nelson models, Comm. Math. Phys. 310 (2012) 511–536. Crossref, Web of Science, ADS, Google Scholar
2. Z. Ammari , Asymptotic completeness for a renormalized nonrelativistic Hamiltonian in quantum field theory: The Nelson model, Math. Phys. Anal. Geom. 3(3) (2000) 217–285. Crossref, Google Scholar
3. Z. Ammari and M. Falconi , Wigner measures approach to the classical limit of the Nelson model: Convergence of dynamics and ground state energy, J. Statist. Phys. 157(2) (2014) 330–362. Crossref, Web of Science, ADS, Google Scholar
4. Z. Ammari and M. Falconi , Bohr’s correspondence principle for the renormalized Nelson model, SIAM J. Math. Anal. 49(6) (2017) 5031–5095. Crossref, Web of Science, Google Scholar
5. Z. Ammari, M. Falconi and F. Hiroshima, Towards a derivation of classical electrodynamics of charges and fields from QED, preprint (2022), arXiv:2202.05015. Google Scholar
6. A. Arai , Ground state of the massless Nelson model without infrared cutoff in a non-Fock representation, Rev. Math. Phys. 13(9) (2001) 1075–1094. Link, Web of Science, Google Scholar
7. V. Betz, F. Hiroshima, J. Lőrinczi, R. A. Minlos and H. Spohn , Ground state properties of the Nelson Hamiltonian: A Gibbs measure-based approach, Rev. Math. Phys. 14(2) (2002) 173–198. Link, Web of Science, Google Scholar
8. C. Boccato, S. Cenatiempo and B. Schlein , Quantum many-body fluctuations around nonlinear Schrödinger dynamics, Ann. Henri Poincaré 18(1) (2016) 113–191. Crossref, Web of Science, ADS, Google Scholar
9. L. Boßmann, S. Petrat, P. Pickl and A. Soffer , Beyond Bogoliubov dynamics, Pure Appl. Anal. 3(4) (2021) 677–726. Crossref, Google Scholar
10. L. Boßmann, S. Petrat and R. Seiringer , Asymptotic expansion of low-energy excitations for weakly interacting bosons, Forum Math. Sigma 9 (2021) E28. Crossref, Web of Science, Google Scholar
11. O. Bratteli and D. W. Robinson , Operator Algebras and Quantum-Statistical Mechanics II. Equilibrium States. Models in Quantum-Statistical Mechanics, Texts and Monographs in Physics (Springer-Verlag, New York, 1981). Google Scholar
12. C. Brennecke, P. T. Nam, M. Napiórkowski and B. Schlein , Fluctuations of N-particle quantum dynamics around the nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 36(5) (2019) 1201–1235. Crossref, Web of Science, ADS, Google Scholar
13. J. T. Cannon , Quantum field theoretic properties of a model of Nelson: Domain and eigenvector stability for perturbed linear operators, J. Funct. Anal. 8(1) (1971) 101–152. Crossref, Google Scholar
14. R. Carlone, M. Correggi, M. Falconi and M. Olivieri , Microscopic derivation of time-dependent point interactions, SIAM J. Math. Anal. 53(4) (2021) 4657–4691. Crossref, Web of Science, Google Scholar
15. J. Chong , Dynamics of large boson systems with attractive interaction and a derivation of the cubic focusing NLS equation in $ℝ^{3}$ , J. Math. Phys. 62(4) (2021) 042106. Crossref, Web of Science, Google Scholar
16. J. Colliander, J. Holmer and N. Tzirakis , Low regularity global well-posedness for the Zakharov and Klein–Gordon–Schrödinger systems, Trans. Amer. Math. Soc. 360(9) (2008) 4619–4638. Crossref, Web of Science, Google Scholar
17. M. Correggi and M. Falconi , Effective potentials generated by field interaction in the quasi-classical limit, Ann. Henri Poincaré 19(1) (2018) 189–235. Crossref, Web of Science, ADS, Google Scholar
18. M. Correggi, M. Falconi and M. Olivieri , Quasi-classical dynamics, to appear in J. Eur. Math. Soc. (2022). Crossref, Google Scholar
19. M. Correggi, M. Falconi and M. Olivieri, Ground state properties in the quasi-classical regime, Anal. PDE (2022), in press, preprint (2020), arXiv:2007.09442. Google Scholar
20. E. B. Davies , Particle-boson interactions and the weak coupling limit, J. Math. Phys. 20 (1979) 345–351. Crossref, Web of Science, ADS, Google Scholar
21. J. Dereziński and C. Gérard , Asymptotic completeness in quantum field theory. Massive Pauli–Fierz Hamiltonians, Rev. Math. Phys. 11(4) (1999) 383–450. Link, Web of Science, Google Scholar
22. J. Dereziński and C. Gérard , Mathematics of Quantization and Quantum Fields, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 2013). Crossref, Google Scholar
23. M. Duerinckx and C. Shirley, Cherenkov radiation with massive bosons and quantum friction, preprint (2022), arXiv:2204.00557. Google Scholar
24. M. Falconi , Classical limit of the Nelson model with cutoff, J. Math. Phys. 54(1) (2013) 012303. Crossref, Web of Science, ADS, Google Scholar
25. M. Falconi , Self-adjointness criterion for operators in Fock spaces, Math. Phys. Anal. Geom. 18(1) (2015) 1–18. Crossref, Web of Science, Google Scholar
26. D. Feliciangeli, S. Rademacher and R. Seiringer , Persistence of the spectral gap for the Landau–Pekar equations, Lett. Math. Phys. 111(19) (2021) 1–19. Google Scholar
27. R. L. Frank and Z. Gang , Derivation of an effective evolution equation for a strongly coupled polaron, Anal. PDE 10(2) (2017) 379–422. Crossref, Web of Science, Google Scholar
28. R. L. Frank and B. Schlein , Dynamics of a strongly coupled polaron, Lett. Math. Phys. 104 (2014) 911–929. Crossref, Web of Science, ADS, Google Scholar
29. C. Gérard , On the scattering theory of massless Nelson models. Rev. Math. Phys. 14(11) (2002) 1165–1280. Link, Web of Science, ADS, Google Scholar
30. C. Gérard, F. Hiroshima, A. Panati and A. Suzuki , Infrared problem for the Nelson model on static space-times, Comm. Math. Phys. 308 (2011) 543–566. Crossref, Web of Science, ADS, Google Scholar
31. C. Gérard, F. Hiroshima, A. Panati and A. Suzuki , Removal of UV cutoff for the Nelson model with variable coefficients, Lett. Math. Phys. 101 (2012) 305–322. Crossref, Web of Science, ADS, Google Scholar
32. J. Ginibre, F. Nironi and G. Velo , Partially classical limit of the Nelson model, Ann. Henri Poincaré 7 (2006) 21–43. Crossref, Web of Science, ADS, Google Scholar
33. J. Ginibre and G. Velo , The classical field limit of scattering theory for non-relativistic many-boson systems. I, Comm. Math. Phys. 66(1) (1979) 37–76. Crossref, Web of Science, ADS, Google Scholar
34. J. Ginibre and G. Velo , The classical field limit of scattering theory for non-relativistic many-boson systems. II, Comm. Math. Phys. 68(1) (1979) 45–68. Crossref, Web of Science, ADS, Google Scholar
35. J. Ginibre and G. Velo , The classical field limit of non-relativistic bosons. II. Asymptotic expansions for general potentials, Ann. Inst. H. Poincaré Phys. Théor. 33 (4) (1980) 363–394. Google Scholar
36. J. Ginibre and G. Velo , The classical field limit of nonrelativistic bosons. I. Borel summability for bounded potentials, Ann. Phys. 128(2) (1980) 243–285. Crossref, Web of Science, ADS, Google Scholar
37. M. Griesemer , On the dynamics of polarons in the strong-coupling limit, Rev. Math. Phys. 29(10) (2017) 1750030. Link, Web of Science, Google Scholar
38. M. Griesemer and A. Wünsch , On the domain of the Nelson Hamiltonian, J. Math. Phys. 59(4) (2018) 042111. Crossref, Web of Science, Google Scholar
39. M. Grillakis and M. Machedon , Pair excitations and the mean field approximation of interacting bosons. I, Comm. Math. Phys. 324(2) (2013) 601–636. Crossref, Web of Science, ADS, Google Scholar
40. M. Grillakis and M. Machedon , Pair excitations and the mean field approximation of interacting bosons. II, Comm. Partial Differential Equations 42(2) (2017) 24–67. Crossref, Web of Science, Google Scholar
41. M. Grillakis, M. Machedon and D. Margetis , Second-order corrections to mean field evolution of weakly interacting bosons. I, Comm. Math. Phys. 294(1) (2010) 273–301. Crossref, Web of Science, ADS, Google Scholar
42. M. Grillakis, M. Machedon and D. Margetis , Second-order corrections to mean field evolution of weakly interacting bosons. II, Adv. Math. 228(3) (2011) 1778–1815. Crossref, Web of Science, Google Scholar
43. M. Gubinelli, F. Hiroshima and J. Lőrinczi , Ultraviolet renormalization of the Nelson Hamiltonian through functional integration, J. Funct. Anal. 267(9) (2014) 3125–3153. Crossref, Web of Science, Google Scholar
44. M. Hirokawa , Infrared catastrophe for Nelson’s model — Non-existence of ground state and soft-boson divergence, Publ. Res. Inst. Math. Sci. 42(4) (2006) 897–922. Crossref, Web of Science, Google Scholar
45. F. Hiroshima , Weak coupling limit with a removal of an ultraviolet cutoff for a Hamiltonian of particles interacting with a massive scalar field, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998) 407–423. Link, Web of Science, Google Scholar
46. F. Hiroshima and O. Matte , Ground states and associated path measures in the renormalized Nelson model, Rev. Math. Phys. 34(2) (2022) 2250002. Link, Web of Science, Google Scholar
47. A. Knowles and P. Pickl , Mean-field dynamics: Singular potentials and rate of convergence, Comm. Math. Phys. 298 (2009) 101–138. Crossref, Web of Science, ADS, Google Scholar
48. E. Kuz , Exact evolution versus mean field second-order correction for bosons interacting via short-range two-body potential, Differ. Integral Equ. 30(7/8) (2017) 587–630. Google Scholar
49. J. Lampart and J. Schmidt , On Nelson-type Hamiltonians and abstract boundary conditions, Comm. Math. Phys. 367(2) (2019) 629–663. Crossref, Web of Science, ADS, Google Scholar
50. N. Leopold, Norm approximation for the Fröhlich dynamics in the mean-field regime, preprint (2022), arXiv:2207.01598. Google Scholar
51. N. Leopold, D. Mitrouskas, S. Rademacher, B. Schlein and R. Seiringer , Landau–Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron, Pure Appl. Anal. 3(4) (2021) 653–676. Crossref, Google Scholar
52. N. Leopold, D. Mitrouskas and R. Seiringer , Derivation of the Landau–Pekar equations in a many-body mean-field limit, Arch. Ration. Mech. Anal. 240 (2021) 383–417. Crossref, Web of Science, Google Scholar
53. N. Leopold and S. Petrat , Mean-field dynamics for the Nelson model with fermions, Ann. Henri Poincaré 20(10) (2019) 3471–3508. Crossref, Web of Science, ADS, Google Scholar
54. N. Leopold and P. Pickl , Mean-field limits of particles in interaction with quantized radiation fields, in Macroscopic Limits of Quantum Systems, eds. D. Cadamuro, M. Duell, W. Dybalski and S. Simonella , Springer Proceedings in Mathematics & Statistics, Vol. 270 (Springer, Cham, 2018), pp. 185–214. Crossref, Google Scholar
55. N. Leopold and P. Pickl , Derivation of the Maxwell–Schrödinger equations from the Pauli–Fierz Hamiltonian, SIAM J. Math. Anal. 52(5) (2020) 4900–4936. Crossref, Web of Science, Google Scholar
56. N. Leopold, S. Rademacher, B. Schlein and R. Seiringer , The Landau–Pekar equations: Adiabatic theorem and accuracy, Anal. PDE 14 (2021) 2079–2100. Crossref, Web of Science, Google Scholar
57. M. Lewin, P. T. Nam and B. Schlein , Fluctuations around Hartree states in the mean-field regime, Amer. J. Math. 137(6) (2015) 1613–1650. Crossref, Web of Science, Google Scholar
58. M. Lewin, P. T. Nam, S. Serfaty and J. P. Solovej , Bogoliubov spectrum of interacting Bose gases, Comm. Pure Appl. Math. 68(3) (2015) 413–471. Crossref, Web of Science, Google Scholar
59. O. Matte and J. S. Møller , Feynman–Kac formulas for the ultra-violet renormalized Nelson model, Astérisque 404 (2018) 1–110. Google Scholar
60. A. Michelangeli, P. T. Nam and A. Olgiati , Ground state energy of mixture of Bose gases, Rev. Math. Phys. 31(2) (2019) 1950005. Link, Web of Science, Google Scholar
61. D. Mitrouskas , A note on the Fröhlich dynamics in the strong coupling limit, Lett. Math. Phys. 111 (2021) 1–24. Crossref, Web of Science, Google Scholar
62. D. Mitrouskas, S. Petrat and P. Pickl , Bogoliubov corrections and trace norm convergence for the Hartree dynamics, Rev. Math. Phys. 31(8) (2019) 1950024. Link, Web of Science, Google Scholar
63. J. S. Møller , The translation invariant massive Nelson model: I. The bottom of the spectrum, Ann. Henri Poincaré 6 (2005) 1091–1135. Crossref, Web of Science, ADS, Google Scholar
64. J. S. Møller , On the essential spectrum of the translation invariant Nelson model, in Mathematical Physics of Quantum Mechanics, Lecture Notes in Physics, Vol. 690 (Springer, Berlin, 2006), pp. 179–195. Crossref, Google Scholar
65. P. T. Nam and M. Napiórkowski , A note on the validity of Bogoliubov correction to mean-field dynamics, J. Math. Pures Appl. 108(5) (2017) 662–688. Crossref, Web of Science, Google Scholar
66. P. T. Nam and M. Napiórkowski , Bogoliubov correction to the mean-field dynamics of interacting bosons, Adv. Theor. Math. Phys. 21(3) (2017) 683–738. Crossref, Web of Science, Google Scholar
67. P. T. Nam and M. Napiórkowski , Norm approximation for many-body quantum dynamics and Bogoliubov theory, in Advances in Quantum Mechanics: Contemporary Trends and Open Problems, eds. A. Michelangeli and G. Dell’Antonio , Springer-INdAM Series, Vol. 18 (Springer, Cham, 2017). Crossref, Google Scholar
68. P. T. Nam and M. Napiórkowski , Norm approximation for many-body quantum dynamics: Focusing case in low dimensions, Adv. Math. 350 (2019) 547–587. Crossref, Web of Science, Google Scholar
69. E. Nelson , Interaction of nonrelativistic particles with a quantized scalar field, J. Math. Phys. 5 (1964) 1190–1197. Crossref, Web of Science, ADS, Google Scholar
70. H. Pecher , Some new well-posedness results for the Klein–Gordon–Schrödinger system, Differ. Integral Equ. 25(1/2) (2012) 117–142. Google Scholar
71. S. Petrat, P. Pickl and A. Soffer , Derivation of the Bogoliubov time evolution for a large volume mean-field limit, Ann. Henri Poincaré 21(2) (2020) 461–498. Crossref, Web of Science, ADS, Google Scholar
72. P. Pickl , A simple derivation of mean field limits for quantum systems, Lett. Math. Phys. 97 (2011) 151–164. Crossref, Web of Science, ADS, Google Scholar
73. A. Pizzo , One-particle (improper) states in Nelson’s massless model, Ann. Henri Poincaré 4 (2003) 439–486. Crossref, Web of Science, ADS, Google Scholar
74. A. Pizzo , Scattering of an infraparticle: The one particle sector in Nelson’s massless model, Ann. Henri Poincaré 6(3) (2005) 553–606. Crossref, Web of Science, ADS, Google Scholar
75. M. Reed and B. Simon , Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness (Academic Press, San Diego, 1975). Google Scholar
76. I. Rodnianski and B. Schlein , Quantum fluctuations and rate of convergence towards mean field dynamics, Comm. Math. Phys. 291(1) (2009) 31–61. Crossref, Web of Science, ADS, Google Scholar
77. J. P. Solovej, Many body quantum mechanics, Lecture Notes (2014), http://web. math.ku.dk/ $\sim$ solovej/MANYBODY/mbnotes-ptn-5-3-14.pdf. Google Scholar
78. S. Teufel , Effective N-body dynamics for the massless Nelson model and adiabatic decoupling without spectral gap, Ann. Henri Poincaré 3 (2002) 939–965. Crossref, Web of Science, ADS, Google Scholar