Trotter Product Theorems and Strong Convergence

Trotter product theorems turn a sequence of split evolutions into the evolution generated by a sum. For unbounded operators, the definition of the sum and the topology of convergence are essential parts of the theorem. Trotter Product Formula develops the standard splitting construction and bounded-operator error expansion; this page supplies precise infinite-dimensional hypotheses and a proof that does not assume a bounded commutator.

Required background. Essential Self-Adjointness identifies a uniquely determined unitary generator. Quadratic Forms defines the different form-sum construction.

Unitary products require a specified self-adjoint sum

Use $\hbar=1$ in this page. Let $A$ and $B$ be self-adjoint operators. Suppose $D(A)\cap D(B)$ is dense and $A+B$ is essentially self-adjoint on that intersection. Set $H=\overline{A+B}$. Then

$$\operatorname{s-lim}_{n\to\infty} \left(e^{-itA/n}e^{-itB/n}\right)^n=e^{-itH}, \qquad t\in\mathbb R.$$

The convergence on each fixed Hilbert vector is uniform for $t$ in compact intervals. No common domain is needed to apply the factors themselves: their exponentials are bounded unitaries. The domain hypothesis identifies their limiting generator.

This statement neither supplies an operator-norm convergence rate nor permits a Baker–Campbell–Hausdorff expansion of unbounded commutators. Such conclusions need additional estimates.

A bounded perturbation admits a direct convergence argument

Assume now that $A$ is self-adjoint and $B$ is bounded self-adjoint. Then $H=A+B$ is self-adjoint on $D(A)$. Define

$$F(h)=e^{-ihA}e^{-ihB},\qquad U(h)=e^{-ihH}.$$

On $D(H)=D(A)$, strong differentiation gives

$$\frac{F(h)\psi-\psi}{h}\longrightarrow-iH\psi, \qquad \frac{U(h)\psi-\psi}{h}\longrightarrow-iH\psi.$$

The difference quotients are uniformly bounded as maps from $D(H)$ with its graph norm into $\mathcal H$. For the first one, use

$$F(h)-I=(e^{-ihA}-I)+e^{-ihA}(e^{-ihB}-I)$$

and $\|(e^{-ihA}-I)\psi\|\leq|h|\|A\psi\|$. Boundedness of $B$ makes the $A$ and $H$ graph norms equivalent.

For fixed $\psi\in D(H)$, the orbit $\{U(s)\psi:|s|\leq T\}$ is compact in that graph norm: both $U(s)\psi$ and $HU(s)\psi=U(s)H\psi$ vary continuously. Uniform boundedness of the difference quotients upgrades their convergence to uniform convergence on this compact orbit. Consequently

$$\sup_{|s|\leq T}\|(F(h)-U(h))U(s)\psi\|=o(|h|).$$

The telescoping identity, with $h=t/n$, yields

$$F(h)^n-U(h)^n =\sum_{j=0}^{n-1}F(h)^{n-1-j} (F(h)-U(h))U(h)^j.$$

All outside factors have norm one. Applying the identity to $\psi$ gives an error at most $n\,o(|t|/n)$, tending to zero. Density of $D(H)$ and the uniform norm bounds extend the result to every Hilbert vector. The compact-orbit argument also gives local uniformity in $t$.

This proof illustrates consistency plus stability. It gives convergence without claiming a rate for the unspecified remainder $o(h)$.

Positive-time products can use a form sum instead

Let $A,B\geq0$ be self-adjoint and suppose

$$Q=D(A^{1/2})\cap D(B^{1/2})$$

is dense. The closed nonnegative form

$$q[\psi]=\|A^{1/2}\psi\|^2+\|B^{1/2}\psi\|^2$$

defines a self-adjoint operator $C$, written $A\dotplus B$. Kato’s form-sum product theorem gives

$$\operatorname{s-lim}_{n\to\infty} \left(e^{-tA/n}e^{-tB/n}\right)^n=e^{-tC}, \qquad t\geq0,$$

again locally uniformly on each vector. Semibounded terms can be shifted first.

Here the form-domain intersection, rather than essential self-adjointness of the operator sum, is the construction hypothesis. The two operators agree when an appropriate operator-sum theorem identifies them, but the notation alone does not establish that equality. In particular, substituting imaginary $t$ into this positive-time convergence theorem does not prove the preceding pointwise unitary product theorem under form assumptions alone.

Time slicing inherits the operator conclusion

For $A=-\alpha\Delta$ and bounded real $B=V$, the unitary theorem applies on $H^2(\mathbb R^d)$. Position-space insertion turns each finite product into the standard time-sliced expression, with the free factors interpreted as oscillatory kernels.

The theorem proves convergence on wave packets. Additional kernel estimates are needed for pointwise endpoint formulas. Why Path Integrals Are Subtle explains this distinction, while Feynman–Kac Formula gives a positive-time probabilistic representation.

Exercise: a discontinuous potential can satisfy the theorem

Take $A=-d^2/dx^2$ on $H^2(\mathbb R)$ and $B$ multiplication by $\mathbf1_{[0,\infty)}$. Does the unitary product formula hold? Does a bounded-commutator BCH estimate follow?

Solution

$B$ is a bounded self-adjoint operator of norm one, so the bounded-perturbation theorem and the proof above apply. However, $B$ generally sends a smooth function nonzero at the origin to a discontinuous function outside $H^2$. Thus $AB$ is not defined on all of $D(A)$, and the formal commutator contains distributional boundary terms. A bounded-operator commutator estimate cannot be inferred. Strong product convergence survives without that invalid error argument.

References

• Kato, Tosio, “Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups,” in Topics in Functional Analysis, Advances in Mathematics Supplementary Studies 3, Academic Press, 1978, pp. 185–195 — the form-sum theorem.
• Simon, Barry, “Tosio Kato’s work on non-relativistic quantum mechanics, Part 2,” Bulletin of Mathematical Sciences 9, 1950005 (2019), Section 18, doi:10.1142/S166436071950005X, author’s text — theorem statements and their distinct domain hypotheses.
• Trotter, Hale F., “On the product of semi-groups of operators,” Proceedings of the American Mathematical Society 10, 545–551 (1959), doi:10.1090/S0002-9939-1959-0108732-6 — the original product-limit theorem.