One-Parameter Groups and Combinatorial Physics

Abstract

1 October 27, 2018 11:23 WSPC/Trim Size: 9in x 6in for Proceedings 1param2 2In this communication, we consider the normal ordering of operators of the typewhere a (resp. a + )is a boson annihilation (resp. creation) operator; these satisfy [a, a + ] ≡ aa + − a + a = 1, and for the purposes of this note may be thought of as a ≡ d/dx and a + ≡ x. We discuss the integration of the one-parameter groups e λΩ and their combinatorial by-products. In particular we show how these groups can be realized as groups of substitutions with prefunctions.

Combinatorics and Boson normal ordering: A gentle introduction

“…Section III discusses the methodology which is applicable to a wide set of problems, for example, the optical Kerr medium as in Eq. (32) and the open system described by Eq. (31).…”

Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups

“…differential operators of the first order exactly, is shown (e.g. in [10,21]) to be of a special type. More precisely, the integration of the one-parameter groups generated by such operators involves transformations going on the name of substitutions with prefunctions in combinatorial physics, that is actually Riordan arrays.…”

“…Following [21,20], the integration of the one-parameter group e λ(q(x)∂x+v(x)) [f (x)] can be considered in the first place when the problem involves a pure vector field only (i.e. v ≡ 0).…”

“…(This gradation appears in fact natural when the algebra HW C is represented by the two usual differential operators X and D.) where q is at least continuous, λ ∈ R is sufficiently small and f is an holomorphic function on (0, 1) (see Comments 1, Section 4). Following [19,21,20], Since (x, λ) → s λ (x) is continuous (and even of class C 1 upon its domain and s 0 (x) = x, s λ is as a deformation of the identity. For small values of λ, the operator e λq(x)∂x coincides with the substitution factor f → f • s λ , for the exponential of a derivation (such as λq(x)∂ x ) is an automorphism, i.e.…”

“…This work is mainly motivated by the algebraic introductory survey of Duchamp, Penson and Tollu [20], by the article of Blaziak and Flajolet [7] and Blasiak, Dattoli, Duchamp, Penson and Solomon [6,8,9,19,21] as well as by the whole bunch of recent papers on Riordan arrays and the Riordan group [4,18,32,44], which extensively investigate the topic. Blaziak and Flajolet's paper provides a notably comprehensive and insighful synthetic presentation, especially with respect to the correspondence with combinatorial objects and structures, such as models that involve special numbers such as generalized Stirling numbers, set partitions, permutations, increasing trees, as well as weighted Dyck and Motzkin lattice paths, rook placement, extensions to q-analogues, multivariate frameworks, urn models, etc.…”