Exact treatment of operator difference equations with nonconstant and noncommutative coefficients
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We study a homogeneous linear second-order difference equation with nonconstant and noncommuting operator coefficients in a vector space. We build its exact resolutive formula consisting of the explicit noniterative expression of a generic term of the unknown sequence of vectors. Some nontrivial applications are reported in order to show the usefulness and the broad applicability of the result.
KeywordsCauchy problem Noncommuting operators Operator difference equations
This paper was written to honor Peter Leach, a profound scientist and a delightful person. A.M. thanks him and the organizers for the pleasant and warm atmosphere at Salt Rock, Durban, South Africa, in November 2011 on the occasion of Peter’s 70th birthday. M.A.J. gratefully acknowledges the financial support of the Erwin Schrodinger International Institute for Mathematical Physics, where parts of this work were carried out. The authors thank Professor Andrzej Jamiolkowski for carefully reading the manuscript and for useful comments.