Analytic splitting data with uniform holomorphic neighbourhood, and the exact symmetric splitting map ${\mathrm{\Psi }}_h$

Let $d\ge 1$. Analytic splitting data consists of the following objects.

1. A nonempty open bounded set $U\subseteq {\mathbb{R}}^{2d}$, star-shaped with respect to $0$ (Star-shaped open subsets of Euclidean space), together with a number $\rho >0$ and the uniform holomorphic neighbourhood $U_{\rho }:=D(\rho )=\{\zeta \in {\mathbb{C}}^{2d}:\mathrm{dist}(\zeta ,\bar{U})<\rho \}$ of Complex phase domains, holomorphic maps on ${\mathbb{C}}^m$, and the complexified canonical form and bracket.
2. Functions $A,B:U\to \mathbb{R}$ (written additively as the splitting $H:=A+B$) that are real analytic on $U$ and admit holomorphic extensions $\tilde{A},\tilde{B}:U_{\rho }\to \mathbb{C}$ with $\tilde{A}{\mathrm{\mid }}_U=A$ and $\tilde{B}{\mathrm{\mid }}_U=B$, where holomorphy is the $\mathbb{C}$-linear total-derivative notion of Complex phase domains, holomorphic maps on ${\mathbb{C}}^m$, and the complexified canonical form and bracket. Set $\tilde{H}:=\tilde{A}+\tilde{B}$ on $U_{\rho }$; then $\tilde{H}{\mathrm{\mid }}_U=H$.
3. A constant $M\ge 1$ with $\parallel X_{\tilde{A}}\parallel \le M$ and $\parallel X_{\tilde{B}}\parallel \le M$ pointwise on $U_{\rho }$, where $X_F=J\mathrm{\nabla }F$ is the complex Hamiltonian field of a holomorphic $F$ and the norm is the Euclidean norm of ${\mathbb{C}}^{2d}\cong {\mathbb{R}}^{4d}$ (Complex phase domains, holomorphic maps on ${\mathbb{C}}^m$, and the complexified canonical form and bracket, A phase domain, its canonical one-form and its standard symplectic two-form).
4. A nonempty compact $K\subseteq U$; then $\delta :=\mathrm{dist}(K,{\mathbb{R}}^{2d}\setminus U)>0$ because $K$ is compact, ${\mathbb{R}}^{2d}\setminus U$ is closed and nonempty, and the two sets are disjoint (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, A compact set and a disjoint closed set have a positive norm-distance gap).

For $F\in \{\tilde{A},\tilde{B}\}$ and $Z\in U_{\rho }$, the complex-flow lemma The flow of a bounded holomorphic field on a polydisc: existence in complex time, displacement bound, holomorphy, and the composition law ↗ supplies an open time neighborhood ${\mathrm{\Omega }}_{F,Z}\subseteq \mathbb{C}$ of $0$ and a uniquely determined holomorphic solution $Y_{F,Z}:{\mathrm{\Omega }}_{F,Z}\to U_{\rho }$ with $Y_{F,Z}(0)=Z$ and $Y_{F,Z}^{\mathrm{\prime }}(\tau )=X_F(Y_{F,Z}(\tau ))$. Write ${\phi }_F^t(Z):=Y_{F,Z}(t)$ when $t\in {\mathrm{\Omega }}_{F,Z}$. Any use of a continuation to a larger connected time domain specifies a single-valued solution branch there; the compositions below are defined only where their three local maps or specified continuations can be composed.

The exact splitting map at step $h\in \mathbb{C}$ is the partial map from $U_{\rho }$ to $U_{\rho }$ given by

${\mathrm{\Psi }}_h:={\phi }_{\tilde{A}}^{h\mathrm{/}2}\circ {\phi }_{\tilde{B}}^h\circ {\phi }_{\tilde{A}}^{h\mathrm{/}2},$

defined wherever the composition exists; for $m\in \mathbb{N}$ its $m$-th iterate is ${\mathrm{\Psi }}_h^m:={\mathrm{\Psi }}_h\circ \cdots \circ {\mathrm{\Psi }}_h$ ($m$ factors, ${\mathrm{\Psi }}_h^0=\mathrm{i}\mathrm{d}$).

• Only $\tilde{A}$ and $\tilde{B}$ enter ${\mathrm{\Psi }}_h$; the real functions $A,B$ serve to state the hypotheses and conclusions on $U$. On real points of $U$, the restrictions $\tilde{A}{\mathrm{\mid }}_U=A$ and $\tilde{B}{\mathrm{\mid }}_U=B$ make their coordinate fields real. Whenever the full real-time flow segments stay in $U$, uniqueness identifies the complex local flows on those data with the corresponding real local flows.
• In the mechanical case of The splitting Hamiltonian $H=A+B$, the drift and kick maps, and the velocity-Verlet map ($\tilde{A}=\frac{1}{2}{p}^{\mathsf{T}}{M}^{-1}p$, $\tilde{B}=V(q)$), the two flows ${\phi }_{\tilde{A}}^t$ and ${\phi }_{\tilde{B}}^t$, restricted to real points and real times whose entire affine drift or kick segment remains in $U$, are exactly the drift $D_t$ and the kick $K_t$ by The drift, kick and velocity-Verlet maps are canonical: $(DC{)}^{\mathsf{T}}JDC=J$ and $C^{\ast }\omega =\omega$; hence ${\mathrm{\Psi }}_h=D_{h\mathrm{/}2}\circ K_h\circ D_{h\mathrm{/}2}$ whenever all three full internal segments remain in $U$. Interchanging the analytic labels $\tilde{A},\tilde{B}$ gives ${\mathrm{\Psi }}_h=V_h=K_{h\mathrm{/}2}\circ D_h\circ K_{h\mathrm{/}2}$. For each order the segment condition applies to every factor at its own initial state; endpoint inclusion alone does not ensure it on arbitrary $U$. In Analytic Verlet retains order-two energy accuracy on exponential times a smaller uniform step bound obtained from $K$ and its margin $\delta$ supplies this condition.
• The bound $\parallel X_F\parallel \le M$ on the complex neighbourhood $U_{\rho }$, not merely on $U$, is an essential hypothesis: it is what makes the analytic continuation and the Cauchy estimates on complex domains available. The number $\rho$ and the margin $\delta$ are fixed data; no smallness is required of $\rho$.