Lauritz van Luijk (@lvanluijk.bsky.social)

The Trotter product formula has countless applications from quantum simulation to numerical analysis—but sharp error bounds, especially for unbounded Hamiltonians, are often hard. We prove new state-dependent bounds and apply them to Schrödinger, Dirac and Coulomb dynamics: doi.org/10.1007/s102...

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Convergence Rates for the Trotter Splitting for Unbounded Operators - Foundations of Computational Mathematics We study convergence rates of the Trotter splitting $$\begin{aligned} e^{A+L} = \lim _{n \rightarrow \infty } \Big (e^{L/n} e^{A/n}\Big )^n \end{aligned}$$ e A + L = lim n → ∞ ( e L / n e A / n ) n in the strong operator topology. In the first part, we use complex interpolation theory to treat generators L and A of contraction semigroups on Banach spaces, with L relatively A-bounded. In the second part, we study unitary dynamics on Hilbert spaces and develop a new technique based on the concept of energy constraints. Our results provide a complete picture of the convergence rates for the Trotter splitting for all common types of Schrödinger and Dirac operators, including singular, confining and magnetic vector potentials, as well as molecular many-body Hamiltonians in dimension $$d=3$$ d = 3 . Using the Brezis-Mironescu inequality, we derive convergence rates for the Schrödinger operator with $$V(x)=\pm |x|^{-a}$$ V ( x ) = ± | x | - a potential. In each case, our conditions are fully explicit. https://doi.org/10.1007/s10208-025-09730-w