A Galton-Watson estimate for Dyson series

• URL: http://arxiv.org/abs/2607.03235v1
• Date: Fri, 03 Jul 2026 11:45:05 GMT
• Title: A Galton-Watson estimate for Dyson series
• Abstract summary: We consider the question of convergence of particular series of integrals, which are labeled by rooted trees.<n> Necessary and sufficient criteria for convergence are obtained, together with an explicit expression for the sum.<n>The interest in these series derives from the Dyson series expansion for the perturbation of a free quantum dynamics by a local potential.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 imlies that the perturbed dynamics exists and is unitarily equivalent with the free one.

Related papers

• Weak coupling limit for quantum systems with unbounded weakly commuting system operators [50.24983453990065]
  This work is devoted to a rigorous analysis of the weak coupling limit (WCL) for the reduced dynamics of an open infinite-dimensional quantum system interacting with electromagnetic field or a reservoir formed by Fermi or Bose particles.<n>We derive in the weak coupling limit the reservoir statistics, which is determined by whose terms in the multi-point correlation functions of the reservoir are non-zero in the WCL.<n>We prove that the resulting reduced system dynamics converges to unitary dynamics with a modified Hamiltonian which can be interpreted as a Lamb shift to the original Hamiltonian.
  arXiv  Detail & Related papers  (2025-05-13T05:32:34Z)
• Continuation of the Stieltjes Series to the Large Regime by Finite-part Integration [0.0]
  We devise a prescription to utilize a novel convergent expansion in the strong-asymptotic regime for the Stieltjes integral. The novel expansion makes use of divergent negative-power moments which we treated as Hadamard's finite part integrals.
  arXiv  Detail & Related papers  (2023-02-08T05:54:33Z)
• Decimation technique for open quantum systems: a case study with driven-dissipative bosonic chains [62.997667081978825]
  Unavoidable coupling of quantum systems to external degrees of freedom leads to dissipative (non-unitary) dynamics. We introduce a method to deal with these systems based on the calculation of (dissipative) lattice Green's function. We illustrate the power of this method with several examples of driven-dissipative bosonic chains of increasing complexity.
  arXiv  Detail & Related papers  (2022-02-15T19:00:09Z)
• Conformal field theory from lattice fermions [77.34726150561087]
  We provide a rigorous lattice approximation of conformal field theories given in terms of lattice fermions in 1+1-dimensions. We show how these results lead to explicit error estimates pertaining to the quantum simulation of conformal field theories.
  arXiv  Detail & Related papers  (2021-07-29T08:54:07Z)
• Perturbation theory without power series: iterative construction of non-analytic operator spectra [0.0]
  It is well known that quantum-mechanical perturbation theory often give rise to divergent series that require proper resummation. Here I discuss simple ways in which these divergences can be avoided in the first place.
  arXiv  Detail & Related papers  (2021-05-11T12:21:18Z)
• Exact thermal properties of free-fermionic spin chains [68.8204255655161]
  We focus on spin chain models that admit a description in terms of free fermions. Errors stemming from the ubiquitous approximation are identified in the neighborhood of the critical point at low temperatures.
  arXiv  Detail & Related papers  (2021-03-30T13:15:44Z)
• Determination of the critical exponents in dissipative phase transitions: Coherent anomaly approach [51.819912248960804]
  We propose a generalization of the coherent anomaly method to extract the critical exponents of a phase transition occurring in the steady-state of an open quantum many-body system.
  arXiv  Detail & Related papers  (2021-03-12T13:16:18Z)
• An Exclusion Principle for Sum-Free Quantum Numbers [0.0]
  The sum-free condition for Schur numbers can be read off as a form of exclusion principle for quantum particles. A particular instance of the correlated exclusion principle can be solved explicitly yielding a sequence of quantum numbers that exhibits a fractal structure. The corresponding algebra of creation and annihilation operators can be identified in terms of commutation and anticommutation relations.
  arXiv  Detail & Related papers  (2020-09-30T08:03:00Z)
• Newton series expansion of bosonic operator functions [0.0]
  We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in cases where the Taylor expansion fails.
  arXiv  Detail & Related papers  (2020-08-25T16:11:31Z)
• Operator-algebraic renormalization and wavelets [62.997667081978825]
  We construct the continuum free field as the scaling limit of Hamiltonian lattice systems using wavelet theory. A renormalization group step is determined by the scaling equation identifying lattice observables with the continuum field smeared by compactly supported wavelets.
  arXiv  Detail & Related papers  (2020-02-04T18:04:51Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.