Integral Equations and Operator Theory

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

Generalization of the Trotter-Lie formula

  • Michel L. Lapidus


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.


Arbitrary Number Vector Potential Scalar Potential Product Formula Usual Product 
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  1. 1.
    Brezis, H. :Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations. Contributions to Nonlinear Functional Analysis. E. Zarantonello ed. Acad. Press New York 1971.Google Scholar
  2. 2.
    Brezis, H.:"Opérateurs Maximaux Monotones et semigroupes de contraction dans les espaces de Hilbert". Mathematics Studies; North-Holland, Amsterdam, 1973.Google Scholar
  3. 3.
    Brezis, H. and Kato, T.:Remarks on the Schrödinger operator with singular complex potentials. J. Math. pures et appl., 58 (1979), 137–151.Google Scholar
  4. 4.
    Brezis, H. and Pazy, A.: Convergence and approximation of semigroups of nonlinear operators in Banach spaces. J. Functional. Anal. 9(1972), 63–74.Google Scholar
  5. 5.
    Chernoff, P.R.:Note on product formulas for operators semigroups. J. Functional Anal. 2(1968), 238–242.Google Scholar
  6. 6.
    Chernoff, P.R.: Semigroup product formulas and addition of unbounded operators. Bull. Amer. Math. Soc. 76 (1970), 395–398.Google Scholar
  7. 7.
    Chernoff, P.R.: Nonassociative addition of unbounded operators and a problem of Brezis and Pazy. Michigan Math. J. 20 (1973), 562–563.Google Scholar
  8. 8.
    Chernoff, P.R.: Product formulas, nonlinear semigroups and addition of unbounded operators. Mem. Amer. Math. Soc. 140 (1974), 1–121.Google Scholar
  9. 9.
    Choquet, G.: "Lectures on Analysis", Vol. II, "Representation Theory". Math. Lectures Notes Series. W.A. Benjamin, New York 1969.Google Scholar
  10. 10.
    Chorin, A.J. Hugues, T.J.R. McCracken, M.F. and Marsden, J.E.: Product formulas and numerical algorithms. Commun. Pure Appl. Math. 31(1978), 205–256.Google Scholar
  11. 11.
    Dunford, N. and Schwartz, J.: "Linear Operators", I, "General Theory". II, "Self-adjoint Operators in Hilbert Spaces". Interscience, New York 1963.Google Scholar
  12. 12.
    Hille, E. and Phillips, R.S. :"Functional Analysis and Semigroups". American Mathematical Society Colloquium Publications, (revised ed), Vol. 31. Providence, Rhode Island, 1957.Google Scholar
  13. 13.
    Hoegh-Krohn, R. and Simon, B.: Hypercontractive semigroups and two-dimensional self-coupled Bose fields. J. Functional Anal. 9(1972), 121–180.Google Scholar
  14. 14.
    Kato, T.: Schrodinger operators with singular potentials. Israel J. of Math. 13(1972), 135–148.Google Scholar
  15. 15.
    Kato, T. : A second look at the essential selfajointness of the Schrödinger operators. Physical Reality and Mathematical Description. D. Reidel Publishing Co. (1974), 193–201.Google Scholar
  16. 16.
    Kato, T.: On the Trotter-Lie product formula. Proc. Japan. Acad. 50 (1974), 694–698.Google Scholar
  17. 17.
    Kato, T.:"Perturbation Theory for Linear.Operators". 2nd ed. Springer-Verlag, Berlin and New York, 1976.Google Scholar
  18. 18.
    Kato, T.: On some Schrödinger operators with a singular complex potential. An. Sc. Norm. Pisa. Ser. IV, 5, (1978), 105–114.Google Scholar
  19. 19.
    Kato, T.: Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups. Topics in Functional Analysis. Ad. Math. Suppl. Studies. I. Gohberg and M. Kac eds, Acad. Press, New York. 3(1978), 185–195.Google Scholar
  20. 20.
    Kato, T. and Masuda, K.: Trotter's product formula for nonlinear semigroups generated by the subdifferentials of convex functionals. J. Math. Soc. Japan, 30(1978), 169–178.Google Scholar
  21. 21.
    Lapidus, M.L.: "Généralisation de la Formule de Trotter-Lie. Etude de Quelques Problèmes liés à des Groupes Unitaires". Thèse de Doctorat d'Université, Université Pierre et Marie Curie, Paris VI, 1980.Google Scholar
  22. 22.
    Lapidus, M.L.: Formules de moyenne et de produit pour les résolvantes imaginaires d'opérateurs autoadjoints. C.R. Acad. Sci. Paris, Ser. A, 291(1980), 451–454.Google Scholar
  23. 23.
    Lapidus, M.L.: Généralisation de la formule de Trotter-Lie. C.R. Acad. Sci. Paris, Ser. A. 291(1980), 497–500.Google Scholar
  24. 24.
    Lapidus, M.L.: Perturbation d'un semi-groupe par un groupe unitaire. C.R. Acad. Sci. Paris, Ser. A. 291 (1980), 535–538.Google Scholar
  25. 25.
    Lapidus, M.L. : The problem of the Trotter-Lie formula for unitary groups of operators. Bull. Sci. Math., to appear.Google Scholar
  26. 26.
    Lapidus, M.L. : Perturbation and nonperturbation theorems in semigroup theory. Applications to the Schrödinger operator. Bull. Sci. Math., to appear.Google Scholar
  27. 27.
    Lapidus, M.L. : Unitary groups of operators, Schrödinger equation and Feynman's thought experiment; to be submitted to Commun. Math. Physics.Google Scholar
  28. 28.
    Lions, J.L. and Stampacchia, G.: Variational inequalities.Commun. Pure Appl. Math. 20(1961), 493–519.Google Scholar
  29. 29.
    Moreau, J.J.: Proximité et dualité dans un espace Hilbertien. Bull. Soc. Math. France, 93(1965), 273–299.Google Scholar
  30. 30.
    Nelson, E.: "Topics in Dynamics", "Flows". Princeton Univ. Press, New Jersey, 1970.Google Scholar
  31. 31.
    Reed, M. and Simon, B.: "Methods of Modern Mathematical Physics", Vol. II. "Fourier Analysis, Self-Adjointness". Acad. Press, New York, 1975.Google Scholar
  32. 32.
    Reich, S.: Product formulas, nonlinear semigroups and accretive operators. J. Functional Anal. 36 (1980), 147–168.Google Scholar
  33. 33.
    Reich, S.: Convergence and approximation of nonlinear semigroups. J. Math. Anal. Appl. 76 (1980), 77–83.Google Scholar
  34. 34.
    Segal, I.: Notes towards the construction of nonlinear relativistic quantum fields. III: Properties of the C*-dynamics for a certain class of interactions. Bull. Amer. Math. Soc. 75(1969), 1390–1395.Google Scholar
  35. 35.
    Simon, B.: An abstract Kato's inequality for generators of positivity preserving semigroups. Indiana Univ. Math. J., 26(1977), 1067–1073.Google Scholar
  36. 36.
    Simon, B.: Maximal and minimal Schrödinger forms. J. Operator Theory. I (1979), 37–47.Google Scholar
  37. 37.
    Trotter, H.: On the product of semigroups of operators. Proc. Amer. Math. Soc. 10(1959), 545–551.Google Scholar

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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