On the exponential solution of differential equations for a linear operator

@article{Magnus1954OnTE,
  title={On the exponential solution of differential equations for a linear operator},
  author={Wilhelm Magnus},
  journal={Communications on Pure and Applied Mathematics},
  year={1954},
  volume={7},
  pages={649-673}
}
  • W. Magnus
  • Published 1954
  • Mathematics
  • Communications on Pure and Applied Mathematics
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… Expand
On some identities involving exponentials
The notion of the exponential of a matrix is usually introduced in elementary textbooks on ordinary differential equations when solving a constant coefficients linear system. One can also find thereExpand
Computational Aspects of Some Exponential Identities
The notion of the exponential of a matrix is usually introduced in elementary textbooks on ordinary differential equations when solving a constant coefficients linear system, also providing some ofExpand
ON GLOBAL REPRESENTATIONS OF THE SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS AS A PRODUCT OF EXPONENTIALS
where the ai(t) are scalar functions of t, and the operators Xi are independent of t. It is further required that the Lie algebra 2 generated by the Xi under the commutator product [Xi, Xj] = XiXjExpand
On the Dimension of Certain Graded Lie Algebras Arising in Geometric Integration of Differential Equations
Many discretization methods for differential equations that evolve in Lie groups and homogeneous spaces advance the solution in the underlying Lie algebra. The main expense of computation is theExpand
Hyperbolic equations with coefficients in an enveloping algebra
where the data zg and the solution u(t) lie in some Banach space B, and the Pi are (unbounded) linear operators in B. This problem has previously been studied by one of us [5, 61 when the Pi areExpand
HOPF ALGEBRAS IN DYNAMICAL SYSTEMS THEORY
The theory of exact and of approximate solutions for non-autonomous linear differential equations forms a wide field with strong ties to physics and applied problems. This paper is meant as aExpand
On the solution of linear differential equations in Lie groups
  • A. Iserles, S. P. Nørsett
  • Mathematics, Biology
  • Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
  • 1999
TLDR
The solution of the linear differential equation y′ = a(t)y, y(0) = y0 is represented as an infinite series whose terms are indexed by binary trees, which leads both to a convergence proof and to a constructive computational algorithm. Expand
University of Cambridge Numerical Analysis Reports on the Solution of Linear Diierential Equations in Lie Groups on the Solution of Linear Diierential Equations in Lie Groups
The subject matter of this paper is the solution of the linear diierential equation y 0 = aty, y0 = y0, where y0 2 G, a : R + ! g and g is a Lie algebra of the Lie group G. By building upon anExpand
Lie algebras associated with the exponential solutions of nonautonomous linear differential equations
A new free Lie algebra structure generated by a continuous set of operators is introduced. The Friedrichs’ criterion, which is known for a finite (countable) number of generators, is extended to theExpand
Note on the Algebraic Aspect of the Integration of a System of Ordinary Linear Differential Equations
In this note the Lie algebra generated by the coefficient matrix of a system of ordinary, linear, first‐order differential equations is considered. A systematic discussion, based on some well‐knownExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 11 REFERENCES
An Operator calculus having applications in quantum electrodynamics
An alteration in the notation used to indicate the order of operation of noncommuting quantities is suggested. Instead of the order being defined by the position on the paper, an ordering subscriptExpand
Errata: A Connection between the Baker-Hausdorff formula and a Problem of Burnside
Let F be the free group of k generators and let V be the subgroup of F which is generated by the pth powers of all the elements of F. Then the quotient group B = F/V has the property that X' = 1 inExpand
Treue Darstellung Liescher Ringe.
Ein Liescher Ring £ mit den Elementen , , ... ist hinsichtlich der distributiven Multiplikation durch die Regeln aa = 0, (ab) c + (bc) a + (ca) b = 0 gekennzeichnet. Es folgt ab + ba = (a + b) (a +Expand
Feynman, An operator calculus having applications in quantum ekctrodynamks
  • Physical Review,
  • 1951
...
1
2
...