Psi-Functions and Frequencies
In this appendix we prove the following theorem stated in section 8. In the form presented it is due to [6], but the proof given here is much simpler, and the normalizing constants are explicitly computed. See also [90] for prior results. — For notations
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D Construction of the Psi-Functions In this appendix we prove the following theorem stated in section 8. In the fonn presented it is due to [6], but the proof given here is much simpler, and the nonnalizing constants are explicitly computed. See also [90] for prior results. - For notations we refer to sections 6 and 7.
Theorem D.I. There exists a complex neighbourhood W of q in W there exist entire functions 1/fn, n ::: 1, satisfying
L6 such that for each
for all m ::: 1. These functions depend analytically on ).. and q and admit a product representation
whose complex coefficients a~ depend real analytically on q and satisfy
Ianm -
T
1< ClYml2
m_
m
for all m, locally uniformly on Wand uniformly in n. We prove this theorem with the help of the implicit function theorem. To this end we refonnulate the statement in tenns of a functional equation. In the following, it is convenient to denote a~ as (j~, and to use the fonner symbol for general £2-sequences. Moreover,
throughout this appendix. T. Kappeler et al., KdV & KAM © Springer-Verlag Berlin Heidelberg 2003
212
VIII Psi-Functions and Frequencies
For a = (am)m>l in.e 2 and n ::: 1 define an entire function cfJn(a) by cfJn(a,)..) =
na
m -).. -2-2-·
-L m rr m.,...n
L5
For q in and m > 1 define a linear functional Am(q) on the space of entire functions by Am(q)cfJ
= _1 2rr
r
d)...
cfJ()..)
lrm Vd 2 ().. , q) -
4
Locally, one can choose the contours r m to be independent of q, and one can choose them arbitrarily close to the real interval
so that Am is actually well defined on the space of real analytic functions on the real line. For each n ::: 1 we then consider on.e 2 x the functional equation
L5
where F n
= (F~)m~l with
-1
Fmn(a,q ) -
A:!z(q)cfJn(a), _ an - rn(q),
m "1= n, m=n,
(D.1)
and, for m "1= n,
In fact, each function F~ is defined and real analytic on some complex neighbourwhich is independent of n and m. hood U of .e 2 x We show that under some mild provisions there exists a unique solution an(q) of Fn(a, q) = 0, which is real analytic in q and extends to some complex neighbourhood of independently of n. We then verify that
L5,
L5
a~
= rm + O(y~/m),
and that this solution satisfies
Thus the functions 'tfrn
will have the required properties.
2 n = -cfJn(a ) rrn
D Construction of the Psi-Functions
213
Real Solutions Before constructing real solutions we first establish the proper setting of the functionals Fn. Lemma D.2. For each n 2: 1, equation (D.1) defines a map F n : £2 x
L5 --+ £2
(a, q)
f-+
r(a, q),
which is real analytic and extends analytically to the complex neighbourhood U of £2 x introduced above. Moreover, this neighbourhood U can be chosen so that all F n are locally uniformly bounded on it.
L5
Proof Fix n, and consider F/:, for m i= n. By the definition of cf>n and the product formula for 112 - 4 in Proposition B.1O, cf>n(a, A)
z/ 112(A) -
(D.2)
4
for A near r m with n A _
Sm ( ) -
(_l)m+! 2
-vx-=-xo
n 2 Ji2 a-n - 1
nV
'" If-m
al -
A
(A21 - A)(A21-! - A)
(D.3)
The absolute v
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