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On the exponential solution of differential equations for a linear operator
Author(s) -
Magnus Wilhelm
Publication year - 1954
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160070404
Subject(s) - operator (biology) , exponential function , citation , mathematics , calculus (dental) , algebra over a field , computer science , mathematical analysis , pure mathematics , library science , medicine , gene , chemistry , dentistry , biochemistry , repressor , transcription factor
The present investigation was stimulated by a recent paper of K. 0. Friedrichs 113, who arrived at some purely algebraic problems in connection with the theory of linear operators in quantum mechanics. In particuIar, Friedrichs used a theorem by which the Lie elements in a free associative ring can be characterized. This theorem is proved in Section I1 of the present paper together with some applications which concern the addition theorem of the exponential function for non-commuting variables, the so-called BakerHausdorff formula. Section I contains some algebraic preliminaries. It is of a purely expository character and so is part of Section 111. Otherwise, Section 111 deals with the following problem, also considered by Friedrichs: Let A( t ) be a linear operator depending on a real variable 1. Let Y(t) be a second operator satisfying the differential equation