Appendix C Proof of the Trotter theorem
The Trotter theorem figures prominently throughout the book, inboth the develop-
ment of numerical solvers for ordinary differential equations and inthe derivation of
the Feynman path integral. However, in eqn. (3.10.18), we presented the theorem with-
out proof. Therefore, in this appendix, we outline the proof of thetheorem following
a technique presented by Schulman (1981).
LetPˆandQˆbe linear operators on a general normed vector spaceVˆ, also known
as aBanach space, and letψ∈Vˆ. The Trotter theorem is equivalent to the statement
that there exists a linear operatorRˆonˆVsuch that the difference
Rˆtψ− lim
n→∞(
Pˆt/nQˆt/n)
ψ= 0, (C.1)where 0≤t <∞. Before proceeding, it is useful to introduce the following definition:
Acontraction semigrouponVˆis a family of bounded linear operatorsPˆt, 0≤t <∞,
which are defined everywhere onˆVand constitute a mappingVˆ→Vˆsuch that the
following statements are true:
Pˆ^0 = 1, PˆtPˆs=Pˆt+s, t≥ 0 , s≤∞, (C.2)lim
t→∞
Pˆtψ=ψ, ||Pˆt||≤ 1. (C.3)Here, the norm||Pˆt||is defined to be
||Pˆt||= inf
β∈B{
β| ||Pˆtφ||≤β||φ|| ∀φ∈Vˆ, ||φ||≤ 1}
. (C.4)
LetAˆ,Bˆ, andAˆ+Bˆbe infinitesimal generators of the contraction semigroupsPˆt,Qˆt,
andRˆt, respectively. This means, for example, that the action ofAˆon a vectorψis
Aψˆ = lim
t→ 01
t(
Pˆtψ−ψ)
. (C.5)
Next, lethbe a positive real number. It is straightforward to verify the following
identity for the contraction semigroups:
(
PˆhQˆh− 1
)
ψ=(
Pˆh− 1)
ψ+Pˆh(
Qˆh− 1)
ψ. (C.6)Using the infinitesimal generators allows us to write