LieTrotter

LieTrotter, short for the Lie–Trotter product formula, is a decomposition used in numerical analysis and mathematical physics to approximate the exponential of a sum of two operators A and B by a product of exponentials of A and B separately. It is particularly useful when A and B do not commute, so that exp(A+B) is difficult to compute directly.

For matrices or bounded operators on a finite- or infinite-dimensional space, the Lie–Trotter formula states that exp(t(A+B)) = lim_{n→∞} [exp(tA/n) exp(tB/n)]^n, under suitable conditions. This convergence is in the operator norm for finite-dimensional cases or on appropriate Banach spaces for bounded operators; unbounded operators require more careful domain considerations.

Using a finite n gives a first-order approximation with an error that scales like 1/n. A related approach, known as Strang splitting, achieves second-order accuracy by symmetrizing the factors, e.g., [exp(tA/2n) exp(tB/n) exp(tA/2n)]^n. Higher-order generalizations, the Lie-Trotter–Suzuki decompositions, extend these ideas to even smaller errors.

Applications of LieTrotter decompositions include simulating time evolution in quantum mechanics, solving time-dependent partial differential equations with split operators, and molecular dynamics simulations where the Hamiltonian is decomposed into manageable parts. In operator theory, the formula underpins proofs such as the Trotter–Kato theorem, relating semigroup generators to sums of generators.

Historical notes: The decomposition bears the names of Sophus Lie and Hale D. Trotter, reflecting early observations about the interaction of noncommuting operators. It remains a foundational tool in numerical linear algebra and computational physics due to its simplicity and broad applicability.