THE HISTORICAL DEVELOPMENT OF PSEUDODIFFERENTIAL OPERATORS
DOI:
https://doi.org/10.55640/Keywords:
pseudodifferential operators, partial differential equations, microlocal analysis, Fourier transform, elliptic operators, spectral theoryAbstract
This article presents a comprehensive historical survey of the theory of pseudodifferential operators, focusing on the motivations behind its emergence, the mathematical problems that necessitated its development, and the major contributions made by leading mathematicians from the nineteenth century to the present day. The limitations of classical differential operators in the study of partial differential equations are analyzed as the primary driving force behind the creation of a more general operator framework. Special attention is given to the role of Fourier analysis, microlocal analysis, elliptic theory, asymptotic methods, and spectral theory in shaping pseudodifferential operator theory. The article highlights the foundational works that established the theory and examines its evolution into a central tool of modern mathematical analysis and mathematical physics.
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References
1.Kohn, J.J., Nirenberg, L. An Algebra of Pseudodifferential Operators. Communications on Pure and Applied Mathematics, 1965.
2.Hörmander, L. The Analysis of Linear Partial Differential Operators, Vols. I–IV. Springer, 1983–1985.
3.Eskin, G. Boundary Value Problems for Elliptic Pseudodifferential Equations. American Mathematical Society, 1981.
4.Demidenko, G.V. Asymptotic Methods in the Theory of Partial Differential Equations. Springer, 2004.
5.Shubin, M.A. Pseudodifferential Operators and Spectral Theory. Springer, 2001.
6.Ivrii, V.Ya. Microlocal Analysis and Precise Spectral Asymptotics. Springer, 1998.
7.Taylor, M.E. Pseudodifferential Operators. Princeton University Press, 1981.
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