Stone's Theorem on One-Parameter Unitary Groups

Derives Lemma A25.2. \(G\) is uniformly continuous on the compact interval \([a,b]\): the Heine–Cantor argument of Theorem 11.25 uses only that the target is a metric space, and \(\mathcal{H}\) is one. So given \(\eta>0\) there is \(\delta>0\) with \(\norm{G(t)-G(s)}<\eta\) whenever \(\abs{t-s}<\delta\). Two tagged partitions of mesh \(<\delta\) have a common refinement, and comparing each with the refinement bounds the difference of their Riemann sums by \(2\eta(b-a)\); the sums therefore form a Cauchy net, which converges because \(\mathcal{H}\) is complete (Definition 16.2). Both identities in Equation (A25.2) hold for Riemann sums — the first by linearity of the inner product in its second slot, the second by the triangle inequality — and pass to the limit, the first by continuity of the inner product (Proposition 16.4). The same argument gives \(B\int G=\int BG\) for a bounded \(B\).

Derives Lemma A25.3. The integrand is continuous, by strong continuity of \(U\) and continuity of \(\varphi\), and vanishes outside a compact interval, so Lemma A25.2 defines \(x_{\varphi}\). Using \(U(s)U(t)=U(t+s)\) and the substitution \(t\longmapsto t-s\) — a translation, which the Riemann integral of Lemma A25.2 respects because it acts on tagged partitions by translation —

Hence, for \(s\neq0\),

By the mean value theorem, \([\varphi(t-s)-\varphi(t)]/s =-\varphi'(t-\theta s)\) for some \(\theta=\theta(t,s)\in(0,1)\), and \(\varphi'\) is continuous with compact support, hence uniformly continuous; so \([\varphi(\cdot-s)-\varphi]/s\longrightarrow-\varphi'\) uniformly as \(s\to0\), with all supports inside one fixed compact interval \([a,b]\) of length \(\ell\) once \(\abs{s}\leq1\). The second estimate of Equation (A25.2) and \(\norm{U(t)x}=\norm{x}\) then give

So the limit in Equation (A25.1) exists and equals \(-x_{\varphi'}\); multiplying by \(\ii\hbar\) gives Equation (A25.4).

Derives Proposition A25.4. For \(n\in\N\) define

Each \(\varphi_{n}\) is continuously differentiable — at \(t=\pm1/n\) both \(\varphi_{n}\) and \(\varphi_{n}'\) vanish, so the pieces match — non-negative, supported in \([-1/n,1/n]\), and normalised: substituting \(u=nt\),

a pure number, as it must be: \(\varphi_{n}\) carries the unit \(/\mathrm{s}\) and \(\dd t\) the unit \(\mathrm{s}\).

Fix \(x\in\mathcal{H}\). Since \(\int\varphi_{n}=1\),

by Equation (A25.2) and non-negativity of \(\varphi_{n}\). The right-hand side tends to \(0\) as \(n\to\infty\), by strong continuity of \(U\) at \(t=0\) (Definition 16.64). Each \(x_{\varphi_{n}}\) lies in \(D(H)\) by Lemma A25.3, so \(x\) is a limit of vectors of \(D(H)\).

Derives Lemma A25.5. Fix \(w\in\mathcal{H}\) and let \(g(r)=\braket{w}{U(r)x}\). By Equation (16.46) the curve \(r\longmapsto U(r)x\) is differentiable in norm with derivative \(-\ii\hbar^{-1}U(r)Hx\), so \(g\) is differentiable with \(g'(r)=-\ii\hbar^{-1}\braket{w}{U(r)Hx}\), which is continuous in \(r\). The fundamental theorem of calculus (Theorem 11.43) and the first identity of Equation (A25.2) give

As \(w\) is arbitrary, Equation (9.31) gives Equation (A25.8).

Derives Proposition A25.6. Let \(x_{n}\in D(H)\) with \(x_{n}\longrightarrow x\) and \(Hx_{n}\longrightarrow z\). Each term of Equation (A25.8) converges: the left side to \(U(t)x-x\), since \(U(t)\) is bounded; and the right side to \(-\ii\hbar^{-1}\int_{0}^{t}U(r)z\,\dd r\), because

Hence \(U(t)x-x=-\ii\hbar^{-1}\int_{0}^{t}U(r)z\,\dd r\) for every \(t\). Dividing by \(t\neq0\) and using continuity of \(r\longmapsto U(r)z\) at \(r=0\),

so the limit Equation (A25.1) exists for \(x\) and equals \(-\ii\hbar^{-1}z\) before multiplication by \(\ii\hbar\). That is, \(x\in D(H)\) and \(Hx=z\): the graph is closed.

Derives Proposition A25.7. \(H\) is densely defined (Proposition A25.4) and symmetric (Equation (16.45)), so \(H\subseteq H^{\dagger}\) and the adjoint exists. Let \(\varepsilon>0\) be an energy.

Step 1: \(\ker(H^{\dagger}\mp\ii\varepsilon)=\set{0}\). Let \(y\in D(H^{\dagger})\) with \(H^{\dagger}y=\ii\varepsilon y\), fix \(x\in D(H)\), and put \(f(t)=\braket{y}{U(t)x}\). By Proposition 16.67, \(U(t)x\in D(H)\) and the curve is differentiable, so

where the second equality is Equation (16.50) and the third uses the antilinearity Equation (9.36), which turns \(\ii\varepsilon\) in the first slot into \(-\ii\varepsilon\) and cancels one factor of \(\ii\) against the prefactor. Hence \(f(t)=f(0)\,\ee^{-\varepsilon t/\hbar}\). But \(\abs{f(t)}\leq\norm{y}\norm{U(t)x}=\norm{y}\norm{x}\) for all \(t\in\R\), while \(\ee^{-\varepsilon t/\hbar}\longrightarrow\infty\) as \(t\longrightarrow-\infty\); so \(f(0)=\braket{y}{x}=0\). As \(x\) ranges over the dense set \(D(H)\), \(y=0\). The case \(H^{\dagger}y=-\ii\varepsilon y\) is the same with \(t\to+\infty\).

Step 2: \(\im(H\pm\ii\varepsilon)=\mathcal{H}\). For \(x\in D(H)\), symmetry makes the cross terms cancel:

If \((H+\ii\varepsilon)x_{n}\longrightarrow u\) then, by Equation (A25.10) applied to differences, both \((x_{n})\) and \((Hx_{n})\) are Cauchy; \(H\) is closed (Proposition A25.6), so \(x_{n}\longrightarrow x\in D(H)\) with \((H+\ii\varepsilon)x=u\). The image is therefore closed. It is also dense: exactly as in Equation (16.53), \(y\perp\im(H+\ii\varepsilon)\) means \(\braket{y}{Hx}=\braket{\ii\varepsilon y}{x}\) for all \(x\in D(H)\), i.e. \(y\in\ker(H^{\dagger}-\ii\varepsilon)=\set{0}\) by Step 1. A dense closed subspace is everything, and likewise for \(H-\ii\varepsilon\).

Step 3. Let \(y\in D(H^{\dagger})\). By Step 2 there is \(x\in D(H)\) with \((H-\ii\varepsilon)x=(H^{\dagger}-\ii\varepsilon)y\). Since \(H\subseteq H^{\dagger}\), this reads \((H^{\dagger}-\ii\varepsilon)(y-x)=0\), so \(y-x=0\) by Step 1 and \(y=x\in D(H)\). Hence \(D(H^{\dagger})=D(H)\) and \(H=H^{\dagger}\).

Derives Lemma A25.8. The identity Equation (A25.10) holds for any symmetric \(H\), so \(H\pm\ii\varepsilon\) is injective with \(\norm{(H\pm\ii\varepsilon)x}\geq\varepsilon\norm{x}\); and \(H\) is closed (Proposition 16.73), so, exactly as in Step 2 of Proposition A25.7, the image is closed. It is dense because \(\bigl(\im(H\pm\ii\varepsilon)\bigr)^{\perp} =\ker(H^{\dagger}\mp\ii\varepsilon)=\ker(H\mp\ii\varepsilon) =\set{0}\), the middle equality being \(H=H^{\dagger}\) and the last one Equation (A25.10) again. Hence \(\im(H\pm\ii\varepsilon)=\mathcal{H}\) and \(H\pm\ii\varepsilon\) is a bijection of \(D(H)\) onto \(\mathcal{H}\). Thus \(V\) is defined on all of \(\mathcal{H}\), and by Equation (A25.10)

so \(V\) is isometric; its image is \(\im(H-\ii\varepsilon)=\mathcal{H}\), so it is unitary. Writing \(u=(H+\ii\varepsilon)x\),

which is Equation (A25.12) — both the formula for the inverse and, since \(x\) runs over \(D(H)\) as \(u\) runs over \(\mathcal{H}\), the identification of \(\im(\identity-V)\) with \(D(H)\). If \(Vu=u\) then Equation (A25.13) gives \(x=0\), hence \(u=(H+\ii\varepsilon)x=0\).

Derives Proposition A25.9. The construction of The Spectral Theorem for a Bounded Self-Adjoint Operator is repeated with Laurent polynomials \(q(z)=\sum_{k=-m}^{m}c_{k}z^{k}\) in place of polynomials, \(q(V)=\sum_{k}c_{k}V^{k}\) with \(V^{-1}=V^{\dagger}\). Three steps need a new argument; everything after them — Propositions A24.8 and A24.9 and the construction and uniqueness of the projection-valued measure in The projection-valued measure — uses only that \(\Phi\) is an isometric, multiplicative, conjugation-preserving map on \(C(K)\) for a compact \(K\subseteq\C\), and applies word for word.

(a) The spectrum lies on the circle. \(\norm{V}=1\), so \(\sigma(V)\) lies in the closed unit disc (Proposition 16.52). For \(\abs{z}<1\) write \(V-z\identity=V(\identity-zV^{-1})\); since \(\norm{zV^{-1}}=\abs{z}<1\) the bracket is invertible by the geometric series of Proposition 16.52 and \(V\) is invertible, so \(z\in\rho(V)\).

(b) Spectral mapping. Fix \(\nu\in\C\) and put \(r(z)=z^{m}\bigl(q(z)-\nu\bigr)\), an ordinary polynomial. Then \(r(V)=V^{m}\bigl(q(V)-\nu\identity\bigr)\) with \(V^{m}\) invertible, so \(q(V)-\nu\identity\) is invertible exactly when \(r(V)\) is, i.e. exactly when \(0\notin\sigma(r(V))=r(\sigma(V))\) (Lemma A24.3). Since \(z\neq0\) on \(\sigma(V)\subseteq\mathbb{T}\), \(r(z)=0\) there means \(q(z)=\nu\); hence \(\sigma(q(V))=q(\sigma(V))\).

(c) The norm identity. With \(\bar q(z)=\sum_{k}c_{k}^{\ast}z^{-k}\) one has \(q(V)^{\dagger}=\bar q(V)\), and on \(\mathbb{T}\), where \(z^{-1}=z^{\ast}\), \(\bar q(z)=q(z)^{\ast}\). So \((\bar q\,q)(V)\) is self-adjoint and, by Equation (16.25), Lemma A24.2 and (b),

Laurent polynomials restricted to the compact set \(\sigma(V)\) form a subalgebra of \(C(\sigma(V))\) that contains the constants, is closed under conjugation (by (c)) and separates points (the function \(z\) does), so Theorem A24.5 makes it dense; with Equation (A25.14) this yields the isometric continuous calculus, and the rest is The Spectral Theorem for a Bounded Self-Adjoint Operator unchanged.

Derives Proposition A25.10. Let \(V\) be the Cayley transform Equation (A25.11) and \(F\) its projection-valued measure (Proposition A25.9). Write

so that \((H+\ii\varepsilon)^{-1}=w(V)\) by Equation (A25.12).

\(h\) is real, and \(F(\set{1})=0\). For \(\abs{z}=1\), \(z\neq1\), multiply numerator and denominator by \(1-z^{\ast}\):

so \(h(z)=-2\varepsilon\operatorname{Im}z/\abs{1-z}^{2}\in\R\), with the unit \(\mathrm{J}\). Since \(w\chi_{\set{1}}=0\) identically, multiplicativity gives \(w(V)F(\set{1})=0\); but \(w(V)\) is injective, so \(F(\set{1})=0\).

The measure. Define \(E(\Omega)=F(h^{-1}(\Omega))\) for Borel \(\Omega\subseteq\R\). Preimages respect complements, intersections and countable unions, so \(E\) inherits every clause of Definition 16.58 from \(F\), with \(E(\R)=F(\mathbb{T}\setminus\set{1})=\identity\). By construction \(\mu_{x}\) is the image of \(\mu^{F}_{x}\) under \(h\), so \(\int(g\circ h)\,\dd\mu^{F}_{x}=\int g\,\dd\mu_{x}\) for every Borel \(g\geq0\) or bounded, and \(g(H):=(g\circ h)(V)\) is the calculus claimed.

The domain. A direct computation on \(\mathbb{T}\), using \(\abs{1\pm z}^{2}=2\pm2\operatorname{Re}z\), gives the identity that makes the transport work:

Now \(D(H)=\im w(V)\) by Equation (A25.12), and

Indeed, if \(x=w(V)u\) then \(\mu^{F}_{x}=\abs{w}^{2}\mu^{F}_{u}\) — because \(\braket{x}{F(\Omega)x} =\braket{u}{(\bar w\chi_{\Omega}w)(V)u}\) — so \(\int\abs{w}^{-2}\dd\mu^{F}_{x}=\norm{u}^{2}<\infty\). Conversely, if the integral is finite put \(g_{n}=w^{-1}\chi_{\set{\abs{w}\geq1/n}}\), a bounded Borel function, and \(u_{n}=g_{n}(V)x\); then \(\norm{u_{n}-u_{m}}^{2}=\int\abs{g_{n}-g_{m}}^{2}\dd\mu^{F}_{x} \longrightarrow0\) by dominated convergence (Remark 16.1), the dominating function being \(\abs{w}^{-2}\), and \(w(V)u_{n}=\chi_{\set{\abs{w}\geq1/n}}(V)x \longrightarrow x\) because \(F(\set{w=0})=F(\set{1})=0\). Hence \(x\in\im w(V)\). Combining Equations (A25.18) and (A25.19) with \(\mu^{F}_{x}(\mathbb{T})=\norm{x}^{2}\),

so the two finiteness conditions agree and the first half of Equation (A25.15) holds.

The action. Let \(x=w(V)u\in D(H)\). Then, by Equation (A25.12),

On the other side, put \(h_{n}=h\,\chi_{\set{\abs{h}\leq n}}\), a bounded Borel function, so that \(h_{n}(H)=\int_{\abs{\lambda}\leq n}\lambda \,\dd E\). Since \(h_{n}w\longrightarrow hw\) pointwise on \(\mathbb{T}\setminus\set{1}\) with \(\abs{h_{n}w}\leq\abs{hw}=\abs{1+z}/2\leq1\), dominated convergence gives

because \(h(z)w(z)=(1+z)/2\) by Equation (A25.16). Pairing with \(y\) and passing to the limit gives the second half of Equation (A25.15); the integral converges absolutely because \(\int\abs{\lambda}\,\dd\abs{\mu_{y,x}}\leq\norm{y} \bigl(\int\lambda^{2}\dd\mu_{x}\bigr)^{1/2}\), by Cauchy–Schwarz on the partition sums.

Proof of Theorem A25.1, converse half. Derives Theorem A25.1. Let \(H\) be self-adjoint, \(E\) its projection-valued measure (Proposition A25.10), and for \(t\in\R\) put

which is legitimate because \(\abs{f_{t}}=1\): the exponent \(\lambda t/\hbar\) is dimensionless, \(\lambda\) being an energy.

Unitary, and a group. \(f_{t}\bar f_{t}=1\) and \(f_{t}f_{s}=f_{t+s}\), \(f_{0}=1\), so multiplicativity and the \(\ast\)-property of the calculus give \(U(t)^{\dagger}U(t)=U(t)U(t)^{\dagger}=\identity\) and Equation (16.41).

Strong continuity. For \(x\in\mathcal{H}\),

by dominated convergence (Remark 16.1): the integrand tends to \(0\) pointwise and is bounded by the constant \(4\), which is \(\mu_{x}\)-integrable because \(\mu_{x}(\R)=\norm{x}^{2}\).

The generator is \(H\). Let \(x\in D(H)\). Then

whose integrand tends to \(0\) pointwise and is bounded by \(4\lambda^{2}/\hbar^{2}\), since \(\abs{\ee^{\ii\theta}-1}\leq\abs{\theta}\) for real \(\theta\); that bound is \(\mu_{x}\)-integrable exactly because \(x\in D(H)\) (Equation (A25.15)). Dominated convergence makes Equation (A25.23) tend to \(0\), so \(x\) lies in the domain Equation (A25.1) of the generator \(G\) of \(U\), with \(Gx=Hx\). Thus \(H\subseteq G\). But \(G\) is self-adjoint by Proposition A25.7 and \(H\) is self-adjoint by hypothesis, so

the middle inclusion because taking adjoints reverses inclusions. Hence \(G=H\): a self-adjoint operator has no proper symmetric extension.

Derives Proposition A25.11. Fix \(x\in D(H)\) and \(t\), and set \(g(s)=W(t-s)U(s)x\) for \(s\in[0,t]\). By Proposition 16.67, \(U(s)x\in D(H)\) and both curves are differentiable, so the product rule — legitimate because each factor is differentiable in norm and \(W\) is isometric — gives

since \(H\) commutes with \(W(t-s)\) on \(D(H)\) (again Proposition 16.67). Hence \(g\) is constant: \(W(t)x=g(0)=g(t)=U(t)x\). As \(D(H)\) is dense (Proposition A25.4) and both operators are bounded, \(U(t)=W(t)\).

For the bijection: Propositions A25.4 and A25.7 show that every group has a self-adjoint generator, and Theorem A25.1 that every self-adjoint operator is the generator of the group Equation (A25.21); the paragraph just proved shows that no group is the exponential of two different self-adjoint operators, since the generator is recovered from the group by Equation (A25.1).