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For a Diophantine frequency vector, the equation $\omega \cdot {\partial }_{x}f=g$ on a periodic torus has an analytic solution on each smaller complex strip exactly when $g$ has zero average. Its nonzero Fourier coefficients are $f_k=g_k/(ik\cdot \omega )$, and its zero coefficient is arbitrary. Thus the normalized zero-mean solution is unique.

For $H=\omega \cdot y+\frac{1}{2}|y{|}^2+ϵV(x)$ with a Diophantine frequency vector, solve ${\mathcal{D}}_{\omega }u=⟨V⟩-V$ and apply translated conjugacy of a Diophantine vector field to $w=ϵ\nabla u$. Set $\eta =ϵ\nabla u+a$ and use the symplectic cotangent lift with a closed momentum shift. The transformed Hamiltonian is exactly $E+\omega \cdot y^{\prime }+\frac{1}{2}{y}^{\prime T}D{\chi }^{-1}D{\chi }^{-T}{y}^{\prime }+\frac{1}{2}|\eta \circ \chi {|}^2$, where $E=ϵ⟨V⟩+\omega \cdot a$. Since $a=O({ϵ}^2)$, the last term is an angular potential of order ${ϵ}^2$ and the quadratic coefficient is identity plus $O(ϵ)$. The construction is on a smaller angular strip and momentum ball.

Take $\mathbb{T}=\mathbb{R}/(2\pi \mathbb{Z})$ and $|k{|}_1={\sum }_j|k_j|$. A Diophantine frequency vector satisfiesChanging the norm only changes the admissible $\gamma$. The bound excludes resonances and quantifies the small divisors in the inverse of ${\mathcal{D}}_{\omega }$.

A periodic solution necessarily has zero-mean right-hand side, because the integral of each angular derivative vanishes. Conversely, if $⟨g⟩=0$, expand its Fourier series:The Diophantine condition makes every denominator nonzero. These are all differentiable periodic solutions: any difference satisfies ${\mathcal{D}}_{\omega }h=0$ and has no nonzero Fourier coefficient. Thus a solution exists exactly for zero-mean $g$, and is unique up to a constant; setting $⟨f⟩=0$ fixes it. The analytic conclusion is on every strictly smaller complex strip, not necessarily the original boundary strip.

Shifting each contour in the Fourier coefficient integral toward the appropriate edge of the complex strip givesHence for $0<\delta <\sigma$,This converges normally, including differentiated series on any still smaller strip, and proves both holomorphic extension and ${\mathcal{D}}_{\omega }f=g$. Real-valuedness follows from $g_{-k}=\overline{g_k}$ and the corresponding identity for $f_k$.

An explicit bound is obtained by counting lattice shells. The number of $k$ with $|k{|}_1=m$ is at mostPut $s=n+\tau$ and $p=s-1\ge 0$. For $p>0$, $m^p{e}^{-\delta m/2}\le (2p/(e\delta ){)}^p$; for $p=0$ use the bound $1$. Since ${\sum }_{m\ge 1}{e}^{-\delta m/2}=(e^{\delta /2}-1{)}^{-1}\le 2/\delta$, this givesUse $0^0=1$ in this constant when $n=1,\tau =0$. The estimate is valid for every positive $\delta$; optimality of the exponent is not needed. For a unit-period flat torus, insert the factors $2\pi$ in both the Fourier exponent and denominator and adjust the constant.