Remarks on the Operator-Norm Convergence of the Trotter Product Formula

Abstract

We revise the operator-norm convergence of the Trotter product formula for a pair {A, B} of generators of semigroups on a Banach space. Operator-norm convergence holds true if the dominating operator A generates a holomorphic contraction semigroup and B is a A-infinitesimally small generator of a contraction semigroup, in particular, if B is a bounded operator. Inspired by studies of evolution semigroups it is shown in the present paper that the operator-norm convergence generally fails even for bounded operators B if A is not a holomorphic generator. Moreover, it is shown that operator norm convergence of the Trotter product formula can be arbitrary slow.

Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces

“…In that case the family {B(t)} t∈I reduces to a non-negative bounded measurable function: b(•) : I −→ R which has to be Hölder continuous with exponent β ∈ (0, 1). For that case it was found in [36] that the convergence rate is O(n −β ) which coincides with (1.7). For the Lipschitz case it was found O(n −1 ) which suggests that (1.8) and (1.9) might be not optimal.…”

“…Proof. [36] immediately yields that the corresponding evolution semigroup {U(τ) = e −τK } τ∈R + satisfies the estimates (3.3) for γ = 1. △ Now we set…”

“…It turns out that the result (1.7) can be hardly improved. Indeed in [36] the simple case H := C and A = 1 was considered. In that case the family {B(t)} t∈I reduces to a non-negative bounded measurable function: b(•) : I −→ R which has to be Hölder continuous with exponent β ∈ (0, 1).…”

Formulas that Represent Cauchy Problem Solution for Momentum and Position Schrödinger Equation

“…An important question is how fast the error decreases in the approximation expression provided by the Chernoff theorem as n tends to infinity, and the same question for the Trotter product formula e A+B = lim n→∞ (e A/n e B/n ) n . The research here is far from the endinig, several recent papers are [64,65,66].…”

Operator-Norm Convergence of the Trotter Product Formula on Hilbert and Banach Spaces: A Short Survey

“…[13, Theorem 3.2]) If the function: q ∈ C 0,β ([0, T ]), β ∈ (0, 1], is non-negative, then for n → ∞ one getssup τ≥0 e −τ(D 0 +Q) − e −τD 0 /n e −τQ/n n = O(1/n β ) .Now a natural question that one may to ask is: what happens, when q is simply continuous?Theorem 4.3 ([13, Theorem 3.3]) If q : [0, 1] → C, is continuous and non-negative, then for n → ∞ e −τ(D 0 +Q) − e −τD 0 /n e −τQ/n n = o(1) . (4.30)…”