Abstract
We consider the question of convergence of particular series of integrals, which are labeled by rooted trees. Necessary and sufficient criteria for convergence are obtained, together with an explicit expression for the sum. The technique used is strongly reminiscent of the generating function approach of Galton and Watson to branching processes. The interest in these series derives from the Dyson series expansion for the perturbation of a free quantum dynamics by a local potential: the convergence of the series implies that the perturbed dynamics exists and is unitarily equivalent with the free one.
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References
Botvich, D., Fayolle, G., Malyshev, V.: Loss networks in thermodinamic limit. In: 11th Conference of Analysis and Optimisation of Systems: Discrete Event Systems. Sopia-Antipolis France 1994. Lecture Notes in Control and Information Science, vol. 199, pp. 465–489. Springer, Berlin (1994)
Botvich, D., Guţă, M., Maassen, H.: Stability of Bose Dynamical Systems and Branching Theory. Preprint (mp-arc 99-130)
Botvich D., Malyshev V.A.: Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi gas. Commun. Math. Phys. 91, 301–312 (1983)
Bergeron F., Labelle G., Leroux P.: Combinatorial Species and Tree-Like Structures. Cambridge University Press, Cambridge (1998)
Caldeira A.C., Leggett A.J.: Influence of dissipation on quantum tunneling in microscopic systems. Phys. Rev. Lett. 46, 211–214 (1981)
Fidaleo F., Liverani C.: Ergodic properties for a quantum non linear dynamics. J. Stat. Phys. 97, 957–1009 (1999)
Fidaleo F., Liverani C.: Ergodic properties of a model related to disordered quantum anharmonic crystals. Commun. Math. Phys. 235, 169–189 (2003)
Ford G., Kac M., Mazur P.: Statistical mechanics of assemblies of coupled oscillators. J. Math. Phys. 6, 504–515 (1965)
Fröhlich J., Merkli M., Ueltchi D.: Dissipative transport: thermal contacts and tunneling junctions. Ann. Henri Poincaré 4, 897–945 (2004)
Harris T.E.: The Theory of Branching Processes. Springer, Berlin (1963)
Jaks̆ić V., Ogata Y., Pillet C.-A.: The Green–Kubo formula for locally interacting fermionic open systems. Ann. Henri Poincaré 8, 1013–1036 (2007)
Joyal A.: Une théorie combinatoire des series formelles. Adv. Math. 42, 1–82 (1981)
Kay B.S.: A uniqueness result in the Segal–Weinless approach to linear Bose fields. J. Math. Phys. 20, 1712–1713 (1979)
Lamb H.: On a peculiarity of the wave-system due to the free vibrations of a nucleus in an extended medium. Proc. Lond. Math. Soc. 2, 88 (1900)
Lewis, J.T., Maassen, H.: Hamiltonian models of classical and quantum stochastic processes. In: Quantum Probability and Applications to the Quantum Theory of Irreversible Processes. Lecture Notes in Mathematics, vol. 1055. Springer, Berlin (1984)
Maassen H.: On the invertibility of Møller morphisms. J. Math. Phys. 23, 1848–1851 (1982)
Maassen H.: Return to equilibrium by a solution of a quantum Langevin equation. J. Stat. Phys. 34, 239–262 (1984)
Robinson D.W.: Return to equilibrium. Commun. Math. Phys. 31, 171–189 (1973)
Spohn H.: Asymptotic completeness for Rayleigh scattering. J. Math. Phys. 38, 2281–2296 (1997)
van Hemmen, J.L.: Dynamics and Ergodicity of the Infinite Harmonic Crystal, a Review of some Salient Features. Springer Lecture Notes in Physics, vol. 93. Springer, Berlin (1979)
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We thank the anonymous referee for corrections and valuable suggestions for improvement.
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Communicated by Claude Alain Pillet.
We dedicate this paper to John T. Lewis, who was a teacher and source of inspiration to both of us.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Maassen, H., Botvich, D. A Galton–Watson Estimate for Dyson Series. Ann. Henri Poincaré 10, 1141–1158 (2009). https://doi.org/10.1007/s00023-009-0014-y
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DOI: https://doi.org/10.1007/s00023-009-0014-y