Summary
We prove that if we extend the algebra of Schwartz distributions in a suitable fashion, the Schrödinger free-particle flow group can be analytically continued for imaginary time to a full group for the diffusion equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Morse:Thermal Physics, second edition (W. A. Benjamin, New York, N.Y., 1969).
G. Wentzel:Quantum Theory of Fields, translation ofEinführung in die Quantentheorie der Wellenfelder (Interscience Publishers, New York, N.Y., 1949), p. 177.
E. Butkov:Mathematical Physics (Addison-Wesley Pub. Co., Reading, Mass., 1986), p. 524.
M. Greenberg:Applications of Green's Function in Science and Engineering (Prentice-Hall Inc., Englewood Cliffs, N.J., 1971).
J. Bjorken andS. Drell:Relativistic Quantum Mechanics (Mc-Graw Hill, New York, N.Y., 1964).
P. Morse andH. Feshbach:Methods of Theoretical Physics, Part 1 (Mc-Graw Hill, New York, N.Y., 1959).
L. Schwartz:Théorie des distributions (Hermann et Cie, Paris, 1951).
I. Halperin:Introduction to the Theory of Distributions (Canadian Mathematical Congress, University of Toronto Press, Toronto, 1952).
G. Temple:Proc. R. Soc. London, Sez. A,228, 175 (1955).
J. P. Marchand:Distributions (North Holland Publ. Comp., Amsterdam, 1962), p. 13.
I. M. Gel'fand et al.:Generalized Functions (Academic Press, New York, N.Y., 1964–68).
M. J. Lighthill:Introduction to Fourier Analysis and Generalised Functions, Cambridge Monographs on Mechanics and Applied Mathematics (Cambridge University Press, Cambridge, 1970).
F. G. Friedlander:Introduction to the Theory of Distributions (Cambridge University Press, New York, N.Y., 1982).
Encyclopedic Dictionary of Mathematics (Mathematical Society of Japan); English translation, edited byK. Ito, second edition, Vol.1, (MIT press, Cambridge, Mass., 1987). p. 477.
A. Erdelyi:Operational Calculus and Generalized Functions (Holt, Rinehart and Winston, New York, N.Y., 1962).
E. Hille andR. Phillips:Functional Analysis and Semi-Groups (American Mathematical Society, 1957).
F. Riesz andB. Sz-Nagy: Translation ofLeçons d'Analyse Fonctionelle (Unger, New York, N.Y., 1955).
S. Chapman andT. Cowling:the Mathematical Theory of Non-uniform Gases; an Account of the Kinetic Theory of Viscosity, Thermal Conduction, and Diffusion in Gases, second edition (Cambridge University Press, Cambridge, 1952).
S. Bochner andW. Martin:Several Complex Variables (Princeton University Press, Princeton, N.J., 1948).
L. Hormänder:Paley-Wiener-Schwartz theorem inThe Analysis of Linear Partial Differential Operators, Vol.1 (Springer-Verlag, Berlin, New York, N.Y., 1983), p. 181.
S. Hawking:A Brief History of Time (Bantam Books, 1988), p. 139.
Author information
Authors and Affiliations
Additional information
The authors of this paper have agreed to not receive the proofs for correction.
Rights and permissions
About this article
Cite this article
Konstantopoulos, C., Mittag, L., Sandri, G. et al. Schrödinger free-particle flow group and the classical diffusion semi-group. Nuov Cim B 106, 1003–1009 (1991). https://doi.org/10.1007/BF02728343
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02728343