29 research outputs found
Exponential Operators, Dobinski Relations and Summability
We investigate properties of exponential operators preserving the particle
number using combinatorial methods developed in order to solve the boson normal
ordering problem. In particular, we apply generalized Dobinski relations and
methods of multivariate Bell polynomials which enable us to understand the
meaning of perturbation-like expansions of exponential operators. Such
expansions, obtained as formal power series, are everywhere divergent but the
Pade summation method is shown to give results which very well agree with exact
solutions got for simplified quantum models of the one mode bosonic systems.Comment: Presented at XIIth Central European Workshop on Quantum Optics,
Bilkent University, Ankara, Turkey, 6-10 June 2005. 4 figures, 6 pages, 10
reference
Bell polynomials in combinatorial Hopf algebras
Partial multivariate Bell polynomials have been defined by E.T. Bell in 1934.
These polynomials have numerous applications in Combinatorics, Analysis,
Algebra, Probabilities, etc. Many of the formulae on Bell polynomials involve
combinatorial objects (set partitions, set partitions in lists, permutations,
etc.). So it seems natural to investigate analogous formulae in some
combinatorial Hopf algebras with bases indexed by these objects. The algebra of
symmetric functions is the most famous example of a combinatorial Hopf algebra.
In a first time, we show that most of the results on Bell polynomials can be
written in terms of symmetric functions and transformations of alphabets. Then,
we show that these results are clearer when stated in other Hopf algebras (this
means that the combinatorial objects appear explicitly in the formulae). We
investigate also the connexion with the Fa{\`a} di Bruno Hopf algebra and the
Lagrange-B{\"u}rmann formula
A product formula and combinatorial field theory
We treat the problem of normally ordering expressions involving the standard boson operators a, ay where [a; ay] = 1. We show that a simple product formula for formal power series | essentially an extension of the Taylor expansion | leads to a double exponential formula which enables a powerful graphical description of the generating functions of the combinatorial sequences associated with such functions | in essence, a combinatorial eld theory. We apply these techniques to some examples related to specic physical Hamiltonians
Some useful combinatorial formulae for bosonic operators
We give a general expression for the normally ordered form of a function
F(w(a,a*)) where w is a function of boson annihilation and creation operators
satisfying [a,a*]=1. The expectation value of this expression in a coherent
state becomes an exact generating function of Feynman-type graphs associated
with the zero-dimensional Quantum Field Theory defined by F(w). This enables
one to enumerate explicitly the graphs of given order in the realm of
combinatorially defined sequences. We give several examples of the use of this
technique, including the applications to Kerr-type and superfluidity-type
hamiltonians.Comment: 8 pages, 3 figures, 17 reference
Bell polynomials in combinatorial Hopf algebras
We introduce partial -Bell polynomials in three combinatorial Hopf
algebras. We prove a factorization formula for the generating functions which
is a consequence of the Zassenhauss formula.Comment: 7 page