Spectral theory of ordinary differential equations
In mathematics the spectral theory of ordinary differential equations is concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. Spectral theory for second order ordinary differential equations on a compact interval was developed by Jacques Charles François Sturm and Joseph Liouville in the ninteenth century and is now known as Sturm-Liouville theory. In modern language it is an application of the spectral theorem for compact operators due to David Hilbert. In his dissertation, published in 1910, Hermann Weyl extended this theory to second order ordinary differential equations with singularities at the endpoints of the interval, now allowed to be infinite or semi-infinite. He simultaneously developed a spectral theory adapted to these special operators and introduced boundary conditions in terms of his celebrated dichotomy between limit points and limit circles.
In the 1920s John von Neumann established a general spectral theorem for unbounded self-adjoint operators, which Kunihiko Kodaira used to streamline Weyl's method. He also generalised Weyl's method to singular ordinary differential equations of even order and obtained a simple formula for the spectral measure. The same formula had also been obtained in independently by E. C. Titchmarsh in 1946 (scientific communication between Japan and the United Kingdom had been interrupted by World War II). Titchmarsh had followed the method of the German mathematician Emil Hilb, who derived the eigenfunction expansions using complex function theory instead of operator theory. Other methods avoiding the spectral theorem were later developed independently by Levitan, Levinson and Yoshida, who used the fact that the resolvant of the singular differential operator could approximated by compact resolvants corresponding to Sturm-Liouville problems for proper subintervals. Another method was found by Mark Grigoryevich Krein; his use of direction functionals was subsequently generalised by I. M. Glazman to arbitrary operators of even order.
Weyl applied his theory to Carl Friedrich Gauss's hypergeometric differential equation, thus obtaining a far-reaching generalisation of the transform formula of F. G. Mehler (1866) for the Legendre differential equation, rediscovered by the Russian physicist Vladimir Fock in 1943, and usually called the "Mehler-Fock transform". This ordinary differential operator is the radial part of the Laplacian operator on 2-dimensional hyperbolic space. More generally, the Plancherel theorem for SL(2,R) of Harish Chandra and Gelfand-Naimark can be deduced from Weyl's theory for the hypergeometric equation, as can the theory of spherical functions for the isometry groups of higher dimensional hyperbolic spaces. Harish Chandra's later development of the Plancherel theorem for general real semisimple Lie groups was strongly influenced by the methods Weyl developed for eigenfunction expansions associated with singular ordinary differential equations. Equally importantly the theory also laid the mathematical foundations for the analysis of the Schrödinger equation and scattering matrix in quantum mechanics.
References
- Akhiezer, Naum Ilich; Glazman, Izrael Markovich (1993), Theory of Linear Operators in Hilbert Space, ISBN 0486677486
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- Coddington, Earl A.; Levinson, Norman (1955), Theory of Ordinary Differential equations, McGraw-Hill, ISBN 0070115427
- Dieudonné, Jean (1988), Treatise on Analysis, Vol. VIII, Academic Press, ISBN 0122155076
- Dunford, Nelson; Schwartz, Jacob T. (1963), Linear Operators, Part II Spectral Theory. Self Adjoint Operators in Hilbert space, Wiley Interscience, ISBN 0471608475
- Hille, Einar (1969), Lectures on Ordinary Differential Equations, Addison-Wesley, ISBN
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- Kodaira, Kunihiko (1949), "The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices", American Journal of Mathematics, 71: 921–945
- Kodaira, Kunihiko (1950), On ordinary differential equations of any even order and the corresponding eigenfunction expansions, vol. 72, pp. 502–544
- Stone, Marshall Harvey (1932), Linear transformations in Hilbert space and Their Applications to Analysis, AMS Colloquium Publications, vol. 16, ISBN 0821810154
- Titchmarsh, Edward Charles (1946), Eigenfunction expansions associated with second order differential equations, Vol. I, first edition, Oxford University Press
- Titchmarsh, Edward Charles (1962), Eigenfunction expansions associated with second order differential equations, Vol. I, second edition, Oxford University Press, ISBN 0608082546
- Vilenkin, Naoum Iakovlevitch (1968), Special Functions and the Theory of Group Representations, Translations of Mathematical Monographs, vol. 22, American Mathematical Society, ISBN 0821815725
- Weidmann, Joachim (1987), Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics, vol. 1258, Springer-Verlag, ISBN 038717902X
- Weyl, Hermann (1910), "Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Functionen", Mathematische Annalen, 68: 220–269
- Weyl, Hermann (1910), "Über gewöhnliche Differentialgleichungen mit Singulären Stellen und ihre Eigenfunktionen", Nachr. Akad. Wiss. Göttingen. Math.-Phys.: 442–446