Semigroup Methods in Mathematical Physics

  • A. T. Bharucha-Reid


The theory of one-parameter semigroups of operators in Banach spaces is concerned with the study of the operator-valued exponential function, its representation and properties, in infinite-dimensional function spaces. The theory can also be considered as a generalization of Stone’s theorem33 on the representation of one-parameter groups of operators in Hilbert space. The theory of semigroups of operators had its origin in 1936, when Hille10 investigated certain concrete semigroups of operators. In 1948, Hille11 published his treatise on semigroups of operators; and in the 1950’s this treatise stimulated a considerable amount of research on semigroups and their applications. In 1957, Hille and Phillips12 published an extensive revision of Hille’s treatise, and at the present time their treatise is the bible of workers in semigroup theory. In addition to the Hille—Phillips treatise, semigroups of operators are the subject of lecture notes by the author2 and Yosida40; chapters on semigroups of operators can be found in the books of Dunford and Schwartz,5 Maurin,20 and Riesz and Sz.Nagy.30 We also refer to the papers of Phillips,27,29 the first of which presents a survey of semigroup theory, while the second gives a very readable introduction to semigroup theory and its applications in the theory of partial differential equations.


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  1. 1.
    A.T. Bharucha-Reid, “Processus de Markov équivalents et semigroups d’opérateurs,” Compt. Rend. Acad. Sci. Paris 257, 1668 (1963).MATHGoogle Scholar
  2. 2.
    A.T. Bharucha-Reid, “Lectures on Semigroups of Operators,” Institute of Mathematical Sciences, Madras, India (1964).MATHGoogle Scholar
  3. 3.
    A.T. Bharucha-Reid, “Note on Markov Processes and Quantum Mechanical Processes.” (To be published.)Google Scholar
  4. 4.
    C.L. Dolph, “Positive Real Resolvents and Linear Passive Hilbert Systems,” Ann. Acad. Sci. Fenn., Ser. A I, No. 336 (1963).Google Scholar
  5. 5.
    N. Dunford and J.T. Schwartz, Linear Operators. Part I. General Theory, Inter-science Publishers, Inc., New York (1958).MATHGoogle Scholar
  6. 6.
    U. Fano, “Description of States in Quantum Mechanics by Density Matrix and Operator Techniques,” Rev. Mod. Phys. 29, 74 (1957).MathSciNetCrossRefADSGoogle Scholar
  7. 7.
    J. Feldman, “On the Schrödinger and Heat Equations for a Non-negative Potential,” Trans. Amer. Math. Soc. 108, 251 (1963).MathSciNetMATHGoogle Scholar
  8. 8.
    W. Feller, “On the Equation of the Vibrating String,” J. Math. Mech. 8, 339, (1959).MathSciNetMATHGoogle Scholar
  9. 9.
    J. Hadamard, Lectures on Cauchy’s Problem in Partial Differential Equations, Yale University Press, New Haven (1923).MATHGoogle Scholar
  10. 10.
    E. Hille, “Notes on Linear Transformations. I,” Trans. Amer. Math. Soc. 39, 131 (1936).MathSciNetMATHGoogle Scholar
  11. 11.
    E. Hille, Functional Analysis and Semigroups, American Mathematical Society, New York (1948).MATHGoogle Scholar
  12. 12.
    E. Hille and R.S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, New York (1957).MATHGoogle Scholar
  13. 13.
    E. Ikenberry, Quantum Mechanics, Oxford University Press, New York (1962).MATHGoogle Scholar
  14. 14.
    T.F. Jordan, M.A. Pinsky, and E.C.G. Sudarshan, “Dynamical Mappings of Density Operators in Quantum Mechanics. II,” J. Math. Phys. 3, 848 (1962).CrossRefADSGoogle Scholar
  15. 15.
    T.F. Jordan and E.C.G. Sudarshan, “Dynamical Mappings of Density Operators in Quantum Mechanics,” J. Math Phys. 2, 772 (1961).MathSciNetCrossRefADSGoogle Scholar
  16. 16.
    T. Kato, “On Linear Differential Equations in Banach Spaces,” Comm. Pure Appl. Math. 9, 479 (1956).MathSciNetCrossRefGoogle Scholar
  17. 17.
    J. Lehner, and G.M. Wing, “On the Spectrum of an Unsymmetric Operator Arising in the Transport Theory of Neutrons,” Comm. Pure. Appl. Math. 8, 217 (1955).MathSciNetCrossRefGoogle Scholar
  18. 18.
    J. Lehner and G.M. Wing, “Solution of the Linearized Boltzmann Equations for the Slab Geometry,” Duke Math. J. 23, 125 (1956).MathSciNetCrossRefGoogle Scholar
  19. 19.
    G.W. Mackey, Mathematical Foundations of Quantum Mechanics, W.A. Benjamin, Inc., New York (1963).MATHGoogle Scholar
  20. 20.
    K. Maurin, Methods of Hilbert Space, Panstwowe Wydawictwo Naukowe, Warsaw (1959).MATHGoogle Scholar
  21. 21.
    J. McConnell, Quantum Particle Dynamics, North Holland Publishing Co., Amsterdam (1960).MATHGoogle Scholar
  22. 22.
    E.W. Montroll, “Markoff Chains, Wiener Integrals, and Quantum Mechanics, Comm. Pure Appl Math. 5, 415 (1952).MathSciNetCrossRefGoogle Scholar
  23. 23.
    T.W. Mulliken, “Semi-Groups of Operators of Class (C0) in Lp Determined by Parabolic Differential Equations,” Pacific J. Math. 9, 791 (1959).MathSciNetCrossRefGoogle Scholar
  24. 24.
    E. Nelson, “Feynman Integrals and the Schrödinger Equation,” J. Math. Phys. 5, 332 (1964).CrossRefADSGoogle Scholar
  25. 25.
    J. Von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton (1955).MATHGoogle Scholar
  26. 26.
    R.S. Phillips, “Perturbation Theory for Semi-Groups of Linear Operators,” Trans. Amer. Math. Soc. 74, 191 (1953).MathSciNetCrossRefGoogle Scholar
  27. 27.
    R.S. Phillips, “Semigroups of Operators,” Bull. Amer. Math. Soc. 61, 16 (1955).MathSciNetCrossRefGoogle Scholar
  28. 28.
    R.S. Phillips. “Dissipative Operators and Hyperbolic Systems of Partial Differential Equations,” Trans. Amer. Math. Soc. 90, 193 (1959).MathSciNetCrossRefGoogle Scholar
  29. 29.
    R.S. Phillips, “Semigroup Methods in the Theory of Partial Differential Equations,” Modern Mathematics for the Engineer, Second Series, McGraw-Hill Book Company, New York (1961), pp. 100–132.Google Scholar
  30. 30.
    F. Riesz, and B. Sz.-Nagy, Lecons d’analyse fonctionelle, Akademia i Kiado, Budapest (1953).MATHGoogle Scholar
  31. 31.
    H. Rubin, “On the Foundations of Quantum Mechanics,” Symposium on the Axiomatic Method, Amsterdam (1959), North Holland Publishing Co., pp. 333–340.Google Scholar
  32. 32.
    I.E. Segal, “Non-Linear Semi-Groups,” Ann Math. 78, 339 (1963).MathSciNetCrossRefGoogle Scholar
  33. 33.
    M.H. Stone, “On One-Parameter Unitary Groups in Hilbert Space,” Ann. Math. 33, 643 (1932).MathSciNetCrossRefGoogle Scholar
  34. 34.
    E.C.G. Sudarshan, Lectures on Foundation of Quantum Mechanics and Field Theory. Madras: Institute of Mathematical Sciences, 1962.Google Scholar
  35. 35.
    B. Sz.-Nagy, “Sur les contractions de l’espace de Hilbert,” Acta Sci. Math. Szeged 15, 87 (1953).MathSciNetMATHGoogle Scholar
  36. 36.
    G.M. Wing, An Introduction to Transport Theory, John Wiley and Sons, New York (1962).Google Scholar
  37. 37.
    K. Yosida, “An Operator-Theoretical Integration of the Wave Equation,” J. Math. Soc. Japan 8, 79 (1956).MathSciNetCrossRefGoogle Scholar
  38. 38.
    K. Yosida, “Integration of the Wave Equation by the Theory of Semi-Groups,” Sugaku 8, 65 (1956–1957).MathSciNetGoogle Scholar
  39. 39.
    K. Yosida, “An Operator-Theoretical Integration of the Temporally Inhomo-geneous Wave Equation,” J. Fac. Sci. Univ. Tokyo, Section 17, 463 (1957).MathSciNetMATHGoogle Scholar
  40. 40.
    K. Yosida, Lectures on Semi-Group Theory and Its Application to Cauchy’s Problem in Partial Differential Equations, Tata Institute of Fundamental Research, Bombay, India (1957).Google Scholar

Copyright information

© Plenum Press 1966

Authors and Affiliations

  • A. T. Bharucha-Reid
    • 1
  1. 1.Wayne State UniversityDetroitUSA

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