Nearly Integrable Infinite-Dimensional Hamiltonian Systems Book

	Introduction
1	Finite-dimensional situation
2	Infinite dimensional systems (the problem and the result)
3	Applications
4	Remarks on averaging theorems
5	Remarks on nearly integrable symplectomorphisms
6	Notations
Pt. 1	Symplectic structures and Hamiltonian systems in scales of Hilbert spaces	1
1.1	Symplectic Hilbert scales and Hamiltonian equations	1
1.2	Canonical transformations	6
1.3	Local solvability of Hamiltonian equations	9
1.4	Toroidal phase space	11
1.5	A version of the former constructions	12
Pt. 2	Statement of the main theorem and its consequences	13
2.1	Statement of the main theorem	14
2.2	Reformulation of Theorem 1.1	18
2.3	Nonlinear Schrodinger equation	22
2.4	Schrodinger equation with random potential	28
2.5	Nonlinear Schrodinger equation on the real line	33
2.6	Nonlinear string equation	35
2.7	On non-commuting operators J, A and partially hyperbolic invariant tori	39
	Appendix. On superposition operator in Sobolev spaces	44
Pt. 3	Proof of the main theorem	45
3.1	Preliminary transformations	45
3.2	Proof of Theorem 1.1	53
3.3	Proof of Lemma 2.2 (solving the homological equations)	67
3.4	Proof of Lemma 3.1 (estimation of the small divisors)	73
3.5	Proof of Lemma 2.3 (estimation of the change of variables)	78
3.6	Proof of Refinement 2	82
3.7	On reducibility of variational equations	84
3.8	Proof of Theorem 1.2	85
	Appendix A. Interpolation theorem	91
	Appendix B. Some estimates for Fourier series	92
	Appendix C. Lipschitz homeomorphisms of Borel sets	92
	Appendix D. Cauchy estimate	93
	List of notations	94
	Bibliography	96
	Index	101