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Open Access Publications from the University of California

Nonlinear Operators and their Propagators

  • Author(s): Schwartz, Charles
  • et al.

Mathematical physicists are familiar with a large set of tools designed for dealing with linear operators, which are so common in both the classical and quantum theories; but many of those tools are useless with nonlinear equations of motion. In this work a general algebra and calculus is developed for working with nonlinear operators: The basic new tool being the ‘‘slash product,’’ defined by A(1+eB) = A+eA/B + O(e^2). For a generic time development equation, the propagator is constructed and then there follows the formal version of time dependent perturbation theory, in remarkable similarity to the linear situation. A nonperturbative approximation scheme capable of producing high accuracy computations, previously developed for linear operators, is shown to be applicable as well in the nonlinear domain. A number of auxiliary mathematical properties and examples are given.

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