Convergence Rates for the Trotter Splitting for Unbounded Operators

TL;DR

This work provides a rigorous, two-pronged analysis of the strong convergence rates for the Trotter splitting with unbounded generators. The first framework leverages Favard spaces and relative $A$-boundedness to obtain $O(1/n)$ and fractional rates for Schrödinger-type operators with singular or mildly coupled perturbations, including molecular Hamiltonians. The second framework introduces energy-constrained, unitary dynamics to derive robust $O(n^{-1})$-type bounds and explicit rates for Schrödinger, harmonic-oscillator, and magnetic Dirac operators, with quantitative results for Coulomb-like and confining potentials. A central technical contribution is the key commutator bound and the stability of regularity across dynamics, enabling state-dependent estimates via Favard spaces and energy norms. The paper also provides concrete applications to Coulomb potentials, hydrogen-like systems, and multi-electron molecules, and backs the theory with numerical experiments that illustrate the predicted convergence behavior. Overall, the results deliver actionable convergence guarantees for splitting-based simulations in quantum dynamics, including challenging unbounded and singular settings.

Abstract

We study convergence rates of the Trotter splitting $e^{A+L} = \lim_{n \to \infty} (e^{L/n} e^{A/n})^n$ 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 $d=3$. Using the Brezis-Mironescu inequality, we derive convergence rates for the Schrödinger operator with $V(x)=\pm |x|^{-a}$ potential. In each case, our conditions are fully explicit.

Figures (4)

• Figure 1: Left: Trotter error for the ground state of respective Schrödinger operator (numerically computed with finite matrix truncation in $\mathbb C^{2M+1}$ of size $M=400$) after time $t=1$. On the right, we see the potential $V_{\alpha}$ for $\alpha \in \{0.51,1,2\}.$
• Figure 2: Trotter error for ground state of respective Schrödinger operator (numerically computed with finite matrix truncation in $\mathbb C^{2M+1}$ of size $M=400$) after time $t=1$ (very non-smooth potential on the left, more regular one in the center). $N$ indicates the number of iterations in the Trotter product. On the right, we show the ground-state wavefunctions of $-\Delta+V_{\alpha}$ for relevant $\alpha.$
• Figure 3: Trotter error for eigenstates (Fock states) $\{1,4,9,14,21,29\}$ of quantum harmonic oscillator (numerically computed with finite matrix truncation in $\mathbb C^{400}$) after time $t=1$. (log-log plot (left)).
• Figure 4: On the left, we see the Trotter error at $t=1$ for all first 81 quantum harmonic oscillator eigenstates (Fock states) after $N=1000$ Trotter iterations (linear scaling) with matrix truncation in $\mathbb C^{400}$. On the right, we see the time dependence of the Trotter error for eigenstates (Fock states) $\{1,4,9,14,21,29\}$.

Theorems & Definitions (67)

• Lemma 2.1: Key commutator bound
• proof : Proof of Lemma \ref{['lem:SemiTrottCommBound']}
• Theorem 2.2: Perturbative $\mathcal{O}(n^{-1})$-Trotter
• Example 2.3: Hölder spaces
• Example 2.4: Besov spaces
• Remark 2.5
• Lemma 2.6
• proof
• Lemma 2.7: Stability of Favard spaces
• proof