References
S.A. Albeverio and R. J. Hoegh Krohn,Mathematical Theory of Feynman Path Integrals, Springer Lecture Notes of Mathematics,523, Springer, Berlin, 1976.
K. Asada, and D. Fujiwara,On the boundedness of integral transformations with rapidly oscillatory kernels, J. Math. Soc. Japan27, (1975), 623–639.
K. Asada and D. Fujiwara,On some oscillatory integral transformations in L 2(R n), Japan. J. Math.4 (1978), 299–361; its summary: Bell. Dept. Elementary Education, Chiba Keizai College, No. 1 (1977), 25–42.
D. G. Babitt,A summation procedure for certain Feynman integrals, J. Math. Phys.4 (1963), 36–41.
G. D. Birkhoff,Quantum mechanics and asymptotic series, Bull. Amer. Math. Soc.39 (1933), 681–700.
A. P. Calderón and R. Vaillancourt,On the boundedness of pseudo-differential operators, J. Math. Soc. Japan23 (1970), 374–378.
R. H. Cameron,A family of integrals serving to connect the Wiener and Feynman integrals, J. Math. Phys.39 (1960), 126–140.
P. Dirac,The Principle of Quantum Mechanics, third ed., Oxford, 1947.
M. V. Fedoryuk,The stationary phase methods and pseudo-differential operators, Russian Math. Surveys26 (1971), 65–115.
R. P. Feynman,Space time approach to nonrelativistic quantum mechanics, Rev. Mod. Phys.20 (1948), 367–387.
R. P. Feynman and A. R. Hibbs,Quantum Mechanics and Path Integral, McGraw-Hill, New York, 1965.
D. Fujiwara,Fundamental solution of partial differential operators of Schrödinger type. I, Proc. Japan Acad.50 (1974), 566–569;II, Proc. Japan Acad.50 (1974), 699–701;III, Proc. Japan Acad.54 (1978), 62–66.
D. Fujiwara,A construction of the fundamental solution for the Schrödinger equation, to appear in Proc. Japan Acad.
D. Fujiwara,A construction of fundamental solution for Schrödinger equation on the spheres, J. Math. Soc. Japan28 (1976), 483–505.
D. Fujiwara,On the boundedness of integral transformations with highly oscillatory kernels, Proc. Japan Acad.51 (1975), 96–99.
I. M. Gelfand and A. M. Yaglom,Integrals in functional spaces and its applications in quantum physics, J. Math. Phys.1 (1960), 48–69.
L. Hörmander,Pseudo-differential operators and hypoelliptic operators, Proc. Symp. on Pure Math. IX, Amer. Math. Soc., 1969.
L. Hörmander,Faurier integral operators, I, Acta Math.127 (1971), 77–183.
L. Hörmander,The existence of wave operators in scattering theory, Math. Z.146 (1976), 69–91.
K. Ito,Generalized uniform complex measures in Hilbertian metric space with their application to the Feynman path integral, Proc. 5th Berkeley Symp. on Mathematical Statistics and Probability, Vol. 2, Univ. of California Press, Berkeley, 1967.
J. B. Keller and D. W. McLaughlin,The Feynman integrals, Amer. Math. Monthly82 (1975), 451–465.
V. P. Maslov,Théorie des perturbations et méthodes asymptotiques, Dunod, Paris, 1970 (French translation)
C. Morette,On the definition and approximation of Feynman path integrals, Phys. Rev.81 (1951), 848–852.
E. Nelson,Feynman path integrals and Schrödinger's equation. J. Math. Phys.5 (1964), 332–343.
H. Rademacher,Uber partielle und totale differenzierbarkeit, I, Math. Ann.99 (1919), 340–359.
J. T. Schwartz,Non-Linear Functional Analysis, Gordon and Breach, New York, 1969.
L. Schwartz,Théorie des distributions, 2c ed., Hermann, Paris, 1966.
H. Whitney,Geometric Integration Theory, Princeton Univ. Press, 1957.
K. Yajima,The quasi-classical limit of quantum scattering theory I, preprint, Univ. of Virginia, 1978;II, Long range scattering, preprint, Univ. of Virginia, 1978.
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Fujiwara, D. A construction of the fundamental solution for the Schrödinger equation. J. Anal. Math. 35, 41–96 (1979). https://doi.org/10.1007/BF02791062
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DOI: https://doi.org/10.1007/BF02791062