Existence, uniqueness and some properties of Schrödinger propagators

  • Kenji Yajima
Part of the Mathematical Physics Studies book series (MPST, volume 12)


We consider a time dependent Schrödinger equation for quantum particles in an electro-magnetic field:
$$ih\frac{{\partial u}}{{\partial t}}\; = H\left( t \right)u = \left( {{H_0}\left( t \right) + V\left( {t,s} \right)} \right)u,\;t \in {R^1},\;x \in {R^n}$$
$$ {H_0}\left( t \right) = \sum\limits_{j = 1}^n {\frac{1}{2}} {\left( {-i\frac{\partial }{{\partial {x_j}}}-{A_j}\left( {t,x} \right)} \right)^2}$$
where V(t, x) and A(t, x) = (A 1(t,x), A 2(t, x),…, A n(t, x)) are the electric scalar and magnetic vector potentials of the field. The purpose of this paper is to report, without proofs, some of the author’s recent results on
  1. (1)

    the existence and uniqueness of a unitary propagator, or fundamental solution, {U (t,s), t,sR 1} in H = L 2 (R n) for (1.1) under most general conditions on A (t, x) and V (t, x); and

  2. (2)

    the regurality and the smoothing properties of the propagator, under various conditions on the potentials.



Fundamental Solution Invariant Subspace Schrodinger Equation Smoothing Effect Weighted Sobolev Space 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    K. Asada and D. Fujiwara, On some oscillatory integral transformation in L 2 (Rn), Japanese J. Math. 4 (1978), 299–361.MathSciNetzbMATHGoogle Scholar
  2. 2.
    L. Carleson, Some analytical problems related to statstical mechanics, “Euclidean Harmonic Analysis” in Lecture Notes in Math. Springer 799 (1979), 5–45.Google Scholar
  3. 3.
    P. Constantin and J. C. Saut, Local smoothing properties of dispersive equations, J. of Amer. Math. Soc. 1 (1988), 413–439.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    H. L. Cycon, R. G. Greose, W. Kirsch and B. Simon, “,” Springer, Berlin-Heidelberg, 1987.zbMATHGoogle Scholar
  5. 5.
    B. E. J. Dahlberg and C. E. Kenig, A note on the almost everywhere behaviour of solution to the Schrödinger equation, Lecture Notes in Math. Springer 908 (1982), 205–208.MathSciNetGoogle Scholar
  6. 6.
    P. A. M. Dirac, “The principle of quantum mechanics,” Oxford Univ. Press, Oxford, 1947.Google Scholar
  7. 7.
    R. E. Edward, “Fourier series, a modern introduction,” Springer, Berlin-New York, 1982.Google Scholar
  8. 8.
    D. Fujiwara, A construction of the fundamental solution for the Schrödinger equation, J. d’Analyse Math. 35 (1979), 41–96.zbMATHCrossRefGoogle Scholar
  9. 9.
    D. Fujiwara, Remarks on convergence of the Feynmann path integrals, Duke Math. J. 47, 559–600.Google Scholar
  10. 10.
    J. Ginibre and G. Velo, The global Cauchy problem for the non-linear Schrödinger equation revisited, ann. Inst. H. Poincaré Anal. Non Linéaire 2, 309–327.Google Scholar
  11. 11.
    J. Goldstein, “Semigroups of linear operators and applications,” Oxford Univ. Press, Oxford, 1985.zbMATHGoogle Scholar
  12. 12.
    J. S. Howland, Stationary scattering theory for time dependent Hamiltonians, Math. Ann. 207 (1974), 315–335.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    W. Hunziker, Distortion analyticity and molecular resonance curves, Ann. l’Inst. H. Poincaré, Phys. Theor. 45 (1986), 339–358.MathSciNetzbMATHGoogle Scholar
  14. 14.
    T. Ikebe and T. Kato, Uniqueness of self-adjoint extension of singular elliptic differential operators, Arch. Rat. Mech. Ana. 9 (1962), 77–92.MathSciNetzbMATHGoogle Scholar
  15. 15.
    T. Kato, Fundamental properties of Hamiltonian operators of Schrödinger type, Trans. Amer. Math. Soc. 70 (1951), 195–211.MathSciNetzbMATHGoogle Scholar
  16. 16.
    T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann. 162 (1966), 258–279.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    T. Kato, Linear evolution equations of “hyperbolic” type, J. Fac. Sci. Univ. Tokyo Sec. I 17 (1972), 241–258.Google Scholar
  18. 18.
    T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, to appear, Reviews in Math. Physics, 2 (1990).Google Scholar
  19. 19.
    C. E. Kenig and A. Ruiz,. A strong type (2, 2) estimatate for a maximal operator associated to the Schrödinger equation, Trans. Amer. Math. Soc. 280 (1983), 239–246.MathSciNetzbMATHGoogle Scholar
  20. 20.
    H. Kitada, On a construction of the fundamental solution for Schrödinger equation, J. Fac. Sci. Univ. Tokyo Sec. IA 27 (1980), 193–226.MathSciNetzbMATHGoogle Scholar
  21. 21.
    H. Kitada and H. Kumanogo, A family of Fourier Integral operators and the fundamental solution for a Schrödinger equation, Osaka J. Math. 18 (1981), 291–360.MathSciNetzbMATHGoogle Scholar
  22. 22.
    K. Masuda, “Evolution equations (in Japanese),” Kinokuniya-Shoten, Tokyo, 1979.Google Scholar
  23. 23.
    T. Matsumura and M. Nagase, On sufficeint conditions for the boundedness of pseuo-differential operators, Proc. Japan Acad. Ser. A 55 (1979), 293–296.CrossRefGoogle Scholar
  24. 24.
    M. Nagase, The L P -boundedness of pseudo-differential operators with non-regular symbols, Comm. P. D. E. 2 (1977), 1045–1061.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    A. Pazy, “A semigroup of linear operators and applications to partial differential equations,” Springer, Berlin-Heidelberg-New York, 1983.Google Scholar
  26. 26.
    M. Reed and B. Simon, “Methods of modern mathematical physics, Vol. II. Fourier ananlysis and selfadjointness,” Academic Press, New York, 1977.Google Scholar
  27. 27.
    P. Sjölin, Regularity of solutions to the Schrödinger equations, Duke Math. J. 55 (1987), 699–715.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    R. Strichartz, Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705–714.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    H. Tanabe, “Evolution equations (in Japanese),” Iwanami-Shoten, Tokyo, 1975.Google Scholar
  30. 30.
    K. Yajima, Existence of solutions for Schrödinger evolution equations, Commun. Math. Phys. 110 (1987), 415–426.MathSciNetADSzbMATHCrossRefGoogle Scholar
  31. 31.
    K. Yajima, On smoothing property of Schrödinger propagators, to appear in Springer Lect. Notes in Math. Proc. of Kato conference.Google Scholar
  32. 32.
    K. Yajima, Schrödinger evolution equations with magnetic field, to appear, J. d’Analyse Math..Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1991

Authors and Affiliations

  • Kenji Yajima
    • 1
  1. 1.Department of Pure and Applied SciencesUniversity of TokyoMeguroku, Tokyo 153Japan

Personalised recommendations