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Sufficiency of H^p-regularity for order‑p operator splittings and explicit error representation

Establish that, for operator splitting methods of order p applied to linear evolution problems generated by L = A + B where A and B may be unbounded, the regularity assumption u ∈ H^p is sufficient to guarantee order p accuracy together with an explicit error representation, thereby confirming that the stronger assumption u ∈ H^{p+1} is not necessary beyond the already demonstrated case p = 2.

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Background

The paper studies operator splitting schemes when the constituent operators may be unbounded, highlighting that classical Taylor-based error analyses can fail due to regularity constraints. For the linear Schrödinger equation and related problems, minimal regularity for classical solutions often conflicts with requirements arising from Taylor expansions.

The authors argue that while order‑p splittings intuitively suggest higher regularity might be needed to bound the error terms, they show for p = 2 that Hp (not H{p+1}) suffices. They conjecture that this phenomenon extends to general p and frame a challenge to provide a proof and an explicit error form under Hp regularity.

References

The minimal conditions consistent with an order-$p$ splitting require $u\in\CC{H}p$ (on the face of it, it might appear that we need $u\in\CC{H}{p+1}$, to ensure that the error term is bounded, but we will see in the sequel that this is not necessary for $p=2$ and it is safe to conjecture that this is the general state of affairs). The challenge is to close the gap between $p$ and $2p$, specifically to prove that $u\in\CC{H}p$, in tandem with formal order conditions, is sufficient for order $p$ {\em and\/} for an explicit form of the error.

— Splitting methods for unbounded operators (2403.15147 - Iserles et al., 22 Mar 2024) in Section 1 (The problem)