Analysis, et Cetera. Research Papers Published in Honor of by Paul H. Rabinowitz, Eduard Zehnder

By Paul H. Rabinowitz, Eduard Zehnder

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It remains to be shown that the theorem holds for some dense set of functions φ in L 2 ( 5 n _ 1 ) . This we show by proving that the theorem holds for all spherical harmonics on 5 n _ 1 . Let St(uj) be any spherical harmonic of degree £ on S' n ~ 1 . We have the following well-known formula (cf. 42) St{u)e(x,u-k)du: = (2π)η/2(-0^(|χ|-1χ)(*|χ|)1-*^+ΐ-ι(*|χ|). 43) 1 Ο^' )) as 0 < r —► oo. 42). 4 holds for the function φ = This completes the proof. 57 St(u). 3 to Bessel functions. Let η be any complex number with \η\ = 1, Im η > 0.

22] Möishezon, B. : On n-dimensional compact varieties with n algebraically independent meromorphic functions, Amer. Math. Soc. Translations, 6 3 (1967), 51-177. , Spieker, B. : Integrability of Dynamical Systems and the Singular-Point Analysis, Foundations of Physics, 15 (6), 1985. , Tabor M. : The Painlevé Property for Partial Differential Equations, J. Math. , 24 (1983), 522-526. : Necessary conditions for the existence of algebraic first integrals, I: Kowalewski's exponents, J. of Celest. , 3 1 (1983), 363-379.

7] Adler, M. : A Systematic Approach Towards Solving Integrable System, preprint 1985, to appear in revised form in Perspective in Mathematics, Academic Press, New York (1989). [8] Adler, M. : T h e Kowalewski and Hénon- Heiles Motions as Manakov Geodesic Flows on SO(4) - a Two Dimensional Family of Lax Pairs, Commun. M a t h . Phys. 1 1 3 , pp. 659-700(1988). [9] Adler, M. : T h e Complex Geometry of the Kowalewski-Painlevé Analysis, to appear in Inventiones M a t h . 1989, preprint May 1988.

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