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Integral Equations and Operator Theory

, Volume 4, Issue 3, pp 366–415 | Cite as

Generalization of the Trotter-Lie formula

  • Michel L. Lapidus
Article

Abstract

Our generalization of the Trotter-Lie formula is based upon the following principle: starting with a "general mean" of semigroups, the corresponding formula must converge to the semigroup generated by an extension of the arithmetic mean of the generators. Accordingly, the usual product formula is associated with a "geometric mean". We establish "average formulæ" for nonlinear semigroups generated by the subdifferentials of convex functionals. We then explore in detail the case of self-adjoint semigroups. A great advantage of these average formulæ is that they naturally hold for an arbitrary number of operators whereas the traditional product formulæ do not usually hold for more than two operators. This fact is important for numerical and theoretical applications. For instance, it enables us to handle with ease the Schrödinger operator withmagnetic vector potential and with arbitrarily singular scalar potential.

Keywords

Arbitrary Number Vector Potential Scalar Potential Product Formula Usual Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag 1981

Authors and Affiliations

  • Michel L. Lapidus
    • 1
  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUnited States

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