Strongly Continuous Unitary Groups

A strongly continuous one-parameter unitary group is a family $\{U(t):t\in\mathbb R\}$ that combines an exact algebraic law with a state-by-state continuity requirement:

$$U(t+s)=U(t)U(s), \qquad U(0)=I, \qquad \lim_{t\to t_0}\|U(t)\psi-U(t_0)\psi\|=0$$

for every $s,t,t_0\in\mathbb R$ and every $\psi\in\mathcal H$. The word strongly names the topology: each vector orbit is norm-continuous. It does not mean that $U(t)$ is continuous in operator norm, and it does not mean that every orbit is differentiable.

This is the correct analytic object for autonomous closed-system evolution and for continuously parametrized symmetries with possibly unbounded generators. The exact correspondence with self-adjoint generators is Stone’s theorem.

Helpful background. State Vectors supplies Hilbert-space norms and unitary evolution. The only additional assumption is the distinction between convergence on each vector and uniform convergence in operator norm.

The group law and strong topology

For each $t$, unitarity means

$$U(t)^*U(t)=U(t)U(t)^*=I.$$

The group law immediately gives

$$U(-t)=U(t)^{-1}=U(t)^*.$$

Strong continuity is continuity in the strong operator topology along this one-parameter family. For fixed $\psi$, the orbit map

$$t\longmapsto U(t)\psi$$

is continuous in the Hilbert-space norm. This quantifier order matters: the same neighborhood of $t_0$ need not work uniformly for every unit vector.

It is enough to check continuity at the identity. Indeed,

$$\begin{aligned} \|U(t)\psi-U(t_0)\psi\| &= \|U(t_0)[U(t-t_0)-I]\psi\|\\ &= \|[U(t-t_0)-I]\psi\|, \end{aligned}$$

so continuity at $0$ implies continuity at every $t_0$. The same calculation also shows that the group acts by isometries on every orbit.

Strong continuity implies continuity of all matrix elements:

$$t\longmapsto\langle\phi,U(t)\psi\rangle.$$

Conversely, for a unitary group, weak continuity of matrix elements implies strong continuity. Taking $\phi=\psi$ gives

$$\|U(t)\psi-\psi\|^2 = 2\|\psi\|^2 - 2\operatorname{Re}\langle\psi,U(t)\psi\rangle,$$

which tends to zero when the relevant matrix element tends to $\langle\psi,\psi\rangle$. This equivalence uses the uniform norm bound $\|U(t)\|=1$; it is not a general equivalence for arbitrary operator families.

Why operator-norm continuity is too restrictive

Operator-norm continuity would require

$$\lim_{t\to0}\|U(t)-I\|=0,$$

where

$$\|U(t)-I\| = \sup_{\|\psi\|=1}\|[U(t)-I]\psi\|.$$

The supremum allows the test vector to depend on $t$. A group with an unbounded generator generally fails this uniform condition even though every fixed state moves continuously. Stone’s theorem sharpens the statement: a strongly continuous unitary group is operator-norm continuous exactly when its self-adjoint generator is bounded.

This distinction is physically appropriate. High-frequency or high-energy vectors can respond rapidly to an arbitrarily small parameter change, while each fixed normalizable state still changes continuously.

Translation on the real line

On $L^2(\mathbb R)$, define

$$(T(a)\psi)(x)=\psi(x-a).$$

Translations are unitary and satisfy

$$T(a)T(b)=T(a+b).$$

They are strongly continuous. One proof begins with $\psi\in C_c(\mathbb R)$, where uniform continuity and compact support imply $\|T(a)\psi-\psi\|_2\to0$. Density of $C_c(\mathbb R)$ in $L^2(\mathbb R)$ and unitarity then extend the result to every $L^2$ vector.

They are not operator-norm continuous. Under the Fourier transform,

$$(\widehat{T(a)\psi})(k)=e^{-ika}\widehat\psi(k).$$

For any $a\ne0$, one can concentrate $\widehat\psi$ near a value where $e^{-ika}=-1$, obtaining vectors for which $\|[T(a)-I]\psi\|$ is arbitrarily close to $2\|\psi\|$. Consequently,

$$\|T(a)-I\|=2, \qquad a\ne0.$$

The strong limit at $a=0$ therefore coexists with a discontinuous operator norm. The eventual generator is momentum divided by $\hbar$, with a proper dense domain.

A diagonal model with an unbounded generator

Let $\mathcal H=\ell^2(\mathbb N_0)$ and define

$$(U(t)c)_n=e^{-int}c_n.$$

The group law and unitarity are immediate. For a fixed $c=(c_0,c_1,\ldots)$,

$$\|U(t)c-c\|^2 = \sum_{n=0}^{\infty} |e^{-int}-1|^2|c_n|^2.$$

Each summand tends to zero and is bounded by $4|c_n|^2$, so dominated convergence proves strong continuity. The generator acts as

$$(Ac)_n=nc_n$$

on the proper dense domain

$$D(A) = \left\{ c\in\ell^2: \sum_{n=0}^{\infty}n^2|c_n|^2<\infty \right\}.$$

This example cleanly separates continuity from differentiability. Every $c\in\ell^2$ has a continuous orbit, but the derivative at $t=0$ exists in norm precisely for $c\in D(A)$.

The generator domain is a derivative domain

With the convention $U(t)=e^{-itA}$, define the candidate generator on

$$D(A) = \left\{ \psi\in\mathcal H: \lim_{t\to0} \frac{U(t)\psi-\psi}{t} \text{ exists in }\mathcal H \right\}$$

by

$$A\psi = i\lim_{t\to0} \frac{U(t)\psi-\psi}{t}.$$

The limit is a strong derivative. It is not an operator-norm derivative, and $D(A)$ is generally not all of $\mathcal H$. Stone’s theorem proves the nontrivial facts that this domain is dense, that $A$ is self-adjoint, and that $U(t)$ is recovered uniquely from the spectral functional calculus of $A$.

For physical time evolution,

$$U(t)=e^{-itH/\hbar},$$

so

$$H\psi = i\hbar \lim_{t\to0} \frac{U(t)\psi-\psi}{t}.$$

The time-dependent Schrödinger equation is therefore a strong derivative statement for vectors in $D(H)$. A vector outside $D(H)$ still evolves unitarily and continuously; it simply need not have a Hilbert-space time derivative.

Groups versus nonautonomous propagators

The one-parameter group law expresses parameter-translation invariance. An autonomous Hamiltonian gives

$$U(t+s)=U(t)U(s).$$

A generic time-dependent Hamiltonian $H(t)$ instead produces a two-parameter propagator $U(t,s)$ satisfying

$$U(t,r)U(r,s)=U(t,s), \qquad U(s,s)=I.$$

Usually $U(t,s)$ depends on both endpoints, not only on $t-s$, so there is no single one-parameter group to which Stone’s theorem can be applied directly. Time ordering, common domains, and existence of the propagator are separate questions.

Likewise, irreversible evolution is often represented by a semigroup defined only for $t\ge0$. A semigroup has no required inverse $U(-t)$ and belongs to a different generator theory. Neither a two-parameter propagator nor a contractive semigroup should be silently called a one-parameter unitary group.

For physical examples of translations, rotations, and conserved quantities, see One-Parameter Unitary Groups.

Exercises

1. A bounded-generator group

Let $B$ be a bounded self-adjoint operator. Show directly from the exponential series that $U(t)=e^{-itB}$ is operator-norm continuous.

Solution

The exponential series converges in operator norm, and

$$e^{-itB}-I = \sum_{n=1}^{\infty} \frac{(-itB)^n}{n!}.$$

Therefore

$$\|e^{-itB}-I\| \leq e^{|t|\|B\|}-1 \longrightarrow0.$$

2. Continuous but not differentiable

For the diagonal group on $\ell^2(\mathbb N_0)$, find a vector with a continuous orbit that does not belong to $D(A)$.

Solution

Choose $c_n=(n+1)^{-1}$ and then normalize. The series $\sum_n|c_n|^2$ converges, so $c\in\ell^2$ and strong continuity applies. But

$$\sum_{n=0}^{\infty}n^2|c_n|^2$$

diverges because its terms approach $1$. Thus the orbit is continuous but has no norm derivative at zero.

3. Group or propagator?

Suppose $H(t)=f(t)H_0$, where $H_0$ is a nonzero self-adjoint operator and $f$ is a real, locally integrable scalar function. The Hamiltonians are scalar multiples of the same operator. Write $U(t,s)$ and determine when it depends only on $t-s$.

Solution

The propagator is

$$U(t,s) = \exp\!\left[ -\frac{i}{\hbar}H_0\int_s^t f(\tau)\,d\tau \right].$$

For nonzero $H_0$, it depends only on $t-s$ for all endpoints exactly when $f$ is constant almost everywhere. To see necessity, restrict to a vector in $D(H_0)$ with $H_0\psi\ne0$ and compare the almost-everywhere time derivatives of a translation-invariant propagator. If $H_0=0$, the evolution would be the identity for every $f$; this is why that case was excluded.

References

• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.
• M. H. Stone, “On one-parameter unitary groups in Hilbert space,” Annals of Mathematics 33, 643–648, 1932, doi:10.2307/1968538.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., American Mathematical Society, 2014.