On the R. A. Sack theorem for translation operators
type: Item
instance of: scholarly article
title: On the R. A. Sack theorem for translation operators
zbMATH Open document ID: 0897.22019
author: O. V. Viskov
published in: Doklady Mathematics
publication date: 1997-08-20T00:00:00Z
review text: In this paper, I suggest an algorithm allowing one to find the ordered form of a polynomial (or of a power series) of the variables \(A\) and \(B\) satisfying the commutation relation \[ [A,B]= AB- BA=aB, \] where \(a\) is a given constant (or a variable commuting with \(A\) and \(B)\). This algorithm, whose statement is given in Section 4 of this paper, generalizes \textit{R. A. Sack}'s theorem [Philos. Mag., VIII. Ser. 3, 497-503 (1958; Zbl 0081.11701), formula (2.11)] asserting that for any polynomial \(F(\cdot)\) (or, at least formally, in a more general situation) the equalities \[ \begin{aligned} F(A+B) & =\exp \bigl\{(B_1/a) \Delta_a \bigr\}\bigl[ F(A) \bigr],\\ F(A+B) & =\exp \bigl\{(B_2/a) \Delta_a\bigr\} \bigl[F(A)\bigr] \end{aligned} \] are valid. Here, \(\Delta_a\) is the finite-difference operator acting on \(F(A)\): \[ \Delta_a \bigl[F(A) \bigr]= F(A+a) -F(A), \] and the subscript in \(B_1\) (respectively, in \(B_2)\) indicates that, when necessary operations are performed in every monomial, the factor containing \(B\) is placed to the left (respectively, to the right) of the factor containing \(A\). The algorithm, stated as Theorems 3 and 4 in Section 4, allows one to find the ordered form \(F(C)\), when the argument \(C\) has the form \(\varphi (B)A+ \psi(B)\) for any functions \(\varphi (B)\) and \(\psi(B)\).
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 47A99
Mathematics Subject Classification ID: 81S05
zbMATH DE Number: 1050683
zbMATH Keywords: polynomial
zbMATH Keywords: finite-difference operator
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1050683