Abstract
In the preceding paper the initial problem to the Schrödinger equations and the Dirac equations were studied with electromagnetic potentials depending on a parameter \(\epsilon\geq0.\) It was proved that if electromagnetic potentials converge as \(\epsilon\rightarrow0\), then so do the solutions to the corresponding equations. In the present paper a generalization of the result on the Dirac equations is given to symmetric hyperbolic systems with coefficients depending continuously on a parameter.
Mathematics Subject Classification (2000). Primary: 35L40.
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References
V.I. Arnold, Ordinary Differential Equations, Nauka, Moscow 1971 (in Russian).
P. Boggiatto, E. Buzano and L. Rodino, Global Hypoellipticity and Spectral Theory, Akademie Verlag, Berlin, 1996.
E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York 1955.
K.O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 (1954), 345–392.
D. Gourdin and T. Gramchev, Global Cauchy problem for hyperbolic systems with characteristics admitting superlinear growth for |x| → ∞, C. R. Acad. Sci. Paris, Ser. I 347 (2009), 49–54.
T. Gramchev, S. Pilipovi´c and L. Rodino, Global regularity and stability in S-spaces for classes of degenerate Shubin operators, in Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations, Operator Theory: Advances and Applications, 205, Birkh¨auser, Basel, 2010, 81–90.
W. Ichinose, A note on the existence and _-dependency of the solution of equations in quantum mechanics, Osaka J. Math. 32 (1995), 327–345.
W. Ichinose, On the Feynman path integral for nonrelativistic quantum electrodynamics, Rev. Math. Phys., to appear. ArXiv:math-ph/0809.4112.
W. Ichinose, The continuity and the differentiability of solutions on parameters to the Schr¨odinger equations and the Dirac equation, preprint.
F. John, Partial Differential Equations, Fourth Edition, Springer-Verlag, New York, 1982.
H. Kumano-go, Pseudo-Differential Operators, MIT Press, Massachusetts, 1981.
S. Lefschetz, Differential Equations: Geometric Theory, Dover, New York, 1977.
S. Mizohata, The Theory of Partial Differential Equations, Cambridge University Press, New York, 1973.
M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Revised and Enlarged Edition, Academic Press, New York, 1980.
M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, Heidelberg, 1987.
M.E. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton
New Jersey, 1981.
B. Thaller, The Dirac Equation, Springer-Verlag, Berlin, Heidelberg, 1992.
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Ichinose, W. (2011). The Continuity of Solutions with Respect to a Parameter to Symmetric Hyperbolic Systems. In: Rodino, L., Wong, M., Zhu, H. (eds) Pseudo-Differential Operators: Analysis, Applications and Computations. Operator Theory: Advances and Applications(), vol 213. Springer, Basel. https://doi.org/10.1007/978-3-0348-0049-5_13
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DOI: https://doi.org/10.1007/978-3-0348-0049-5_13
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