By Berti M., Bolle P.
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Additional info for Cantor families of periodic solutions for completely resonant nonlinear wave equations
By the spectral theory of compact self-adjoint operators in Hilbert spaces, there is a , ε -orthonormal basis (vk,j )j≥1,j=k of Fk , such that vk,j is an eigenvector of Kk (ε) associated to a positive eigenvalue νk,j (ε), the sequence (νk,j (ε))j is non-increasing and tends to 0 as j → +∞. Each vk,j (ε) belongs to D(Sk ) and is an eigenvector of Sk with associated eigenvalue λk,j (ε) = 1/νk,j (ε), with (λk,j (ε))j≥1 → +∞ as j → +∞. The map ε → Kk (ε) ∈ L(Fk , Fk ) is differentiable and Kk (ε) = −Kk (ε)M Kk (ε), where M u := πk (a0 u).
Kuksin, J. P¨ oschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schr¨ odinger equation, Ann. of Math, 2, 143, no. 1, 149-179, 1996.  J. Moser, Periodic orbits near an Equilibrium and a Theorem by Alan Weinstein, Comm. Pure Appl. , vol. XXIX, 1976.  J. P¨ oschel, A KAM-Theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisa, Cl. , 23, 119-148, 1996.  J. P¨ oschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. , 71, no. 2, 269-296, 1996.
As in section 4, setting a(t, x) := (∂u gδ )(x, v(t, x) + δw(t, x)), we can decompose Ln (δ, v1 , y) = D − M1 − M2 where (with the notations of section 4) Dh := Lω h − δPn ΠW (a0 (x)h) M1 h := δPn ΠW (a(t, x)h) M2 h := δPn ΠW Dv Γ(δ, v1 + v2 (δ, v1 , y), y)Dy v2 (δ, v1 , y)[h] . 1, the eigenvalues of the similarly defined operator Sk satisfy λk,j = j 2 + δM (δ, v1 , y) + O(δ/j), where M (δ, v1 , y) := 1 |Ω| (∂u gδ ) x, v1 + v2 (δ, v1 , y) + δw(t, x) dxdt, w = a2 L−1 (v 2 ) + y . 2) still hold assuming an analogous non resonance condition, and we can define in the same way the operators U, R1 , R2 , with U −1 h σ,s = (1 + O(δ)) h σ,s .
Cantor families of periodic solutions for completely resonant nonlinear wave equations by Berti M., Bolle P.