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Title: An exact formulation of the time-ordered exponential using path-sums

We present the path-sum formulation for the time-ordered exponential of a time-dependent matrix. The path-sum formulation gives the time-ordered exponential as a branched continued fraction of finite depth and breadth. The terms of the path-sum have an elementary interpretation as self-avoiding walks and self-avoiding polygons on a graph. Our result is based on a representation of the time-ordered exponential as the inverse of an operator, the mapping of this inverse to sums of walks on a graphs, and the algebraic structure of sets of walks. We give examples demonstrating our approach. We establish a super-exponential decay bound for the magnitude of the entries of the time-ordered exponential of sparse matrices. We give explicit results for matrices with commonly encountered sparse structures.
Authors:
 [1] ;  [2] ;  [3] ;  [1] ;  [4]
  1. Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU (United Kingdom)
  2. Colby College, 4000 Mayflower Hill Dr., Waterville, Maine 04901 (United States)
  3. Department of Physics and Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 Munich (Germany)
  4. (Singapore)
Publication Date:
OSTI Identifier:
22403149
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONTINUED FRACTIONS; DIAGRAMS; GRAPH THEORY; MATRICES; TIME DEPENDENCE