Error bounds on exponential product formulas for Schroedinger operators

摘要

We study an error bound in the operator norm for the Trotter-Kato product formula of Schroedinger semigroups with potentials growing at infinity. Let H = -Δ + V = H_0 + V be the Schroedinger operator acting on the space L~2 = L~2(R~n). Then the Trotter-Kato product formula says that s - lim_(N→∞) K(t/N)~N = exp(-tH) strongly in L~2, where K(t) : L~2→ L~2 is denned as K(t) = exp(-tV/2)exp(-tH_0)exp(-tV/2), t ≥ 0, (1.1) which is often called the Kac operator or the transfer operator in statistical mechanics. The aim here is to evaluate the error bound in the operator norm as N → ∞ for the exponential product formula above.