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Stationary rays of a self-adjoint flow are exactly the eigenrays

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H \neq {0}$ be a complex Hilbert space (Hilbert space, Real and complex inner product spaces, with the inner product linear in the first argument), let $T$ be self-adjoint (Symmetric, self-adjoint and essentially self-adjoint operators) with spectral projection-valued measure $E$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure), let $U (t) = e^{i t T} = \int e^{i t λ} d E (λ)$ be the generated strongly continuous unitary group (Strongly continuous one-parameter unitary group, Integral of a measurable function against a projection-valued measure), and let $ψ \in H$ with $ψ \neq 0$ . Then the following are equivalent:

(i) for every $t \in R$ the vector $U (t) ψ$ is a complex multiple of $ψ$ (the ray of $ψ$ is invariant under the flow);

(ii) $ψ \in D (T)$ and $T ψ = λ ψ$ for some $λ \in R$ (the ray of $ψ$ is an eigenray of $T$ ).

If these hold for the eigenvalue $λ$ , then $U (t) ψ = e^{i t λ} ψ$ for every $t \in R$ .

Facts & Assumptions

Given: The Axiom of Choice and the Axiom of Countable Choice; a nonzero complex Hilbert space $H$ ; a self-adjoint operator $T$ with spectral PVM $E$ and flow $U (t) = \int e^{i t λ} d E (λ)$ ; a nonzero vector $ψ \in H$ .

[F1]

Flow and domain. $U$ is a strongly continuous one-parameter unitary group, $U (t) D (T) = D (T)$ , and for $u \in D (T)$ the difference quotient $\frac{1}{s} (U (s) u - u)$ converges to $i T u$ as $s \to 0$ , while for $u \notin D (T)$ it does not converge; moreover $E (B) U (t) = U (t) E (B)$ for every Borel $B$ (Flow calculus of a self-adjoint generator: commutation, invariance and differentiability clause (a)-(c)).

[F2]

Flow as a bounded function of $T$ . $U (t) = Φ_{E} (f_{t})$ with $f_{t} (λ) = e^{i t λ}$ , and for $x \in H$ and bounded Borel $g$ one has $∥ Φ_{E} (g) x ∥^{2} = \int ∣ g ∣^{2} d E_{x}$ , where $E_{x} (B) = ⟨ E (B) x, x ⟩$ (Bounded borel pvm integral clause 2, The unbounded PVM integral is densely defined, closed and normal).

[F3]

Eigenvector identity. For $u \in D (T)$ and real $μ$ one has $∥ (T - μ) u ∥^{2} = \int (λ - μ)^{2} d E_{u} (λ)$ (The unbounded PVM integral is densely defined, closed and normal).

[F4]

Vanishing criterion. If $g \geq 0$ is measurable with $\int g d E_{u} = 0$ , then $g = 0$ $E_{u}$ -almost everywhere (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

[F5]

Closed subspaces and Stone's theorem. The span $K = C u$ of a nonzero vector is closed (A finite-dimensional normed subspace is closed) and, with the restricted inner product, is a complex Hilbert space (Hilbert space, A closed subspace of a Banach space is Banach); a strongly continuous one-parameter unitary group on a complex Hilbert space equals $t \mapsto e^{i t S}$ for a unique self-adjoint generator $S$ , and $D (S)$ is exactly the set of vectors whose difference quotients converge, with limit $i S x$ (Stone's theorem: unitary groups and self-adjoint generators, Strongly continuous one-parameter unitary group).

[F6]

Reality of eigenvalues. A self-adjoint operator is symmetric: $⟨ S u, u ⟩ = ⟨ u, S u ⟩$ for every $u \in D (S)$ (Symmetric, self-adjoint and essentially self-adjoint operators); if $S u = λ u$ with $u \neq 0$ and $u \in D (S)$ , then $λ ∥ u ∥^{2} = ⟨ S u, u ⟩ = ⟨ u, S u ⟩ = \overline{λ} ∥ u ∥^{2}$ , so $λ \in R$ (Real and complex inner product spaces, with the inner product linear in the first argument).

Proof

technique · direct; the forward implication reads the spectral measure of an eigenvector, the converse restricts the flow to the closed invariant span of $\psi$ and applies Stone's theorem there

Given: The data of the statement, and [F1]-[F6].

1.1givenF2F3F4F6algebra

[Eigenrays are stationary] Assume (ii): $T ψ = λ ψ$ with $λ \in R$ , and set $u : = ψ / ∥ ψ ∥$ , so that $T u = λ u$ and $∥ u ∥ = 1$ . By [F3], $\int (λ^{'} - λ)^{2} d E_{u} (λ^{'}) = ∥ (T - λ) u ∥^{2} = 0$ ; the integrand is nonnegative and measurable, so by [F4] it vanishes $E_{u}$ -almost everywhere, that is, $E_{u} (R ∖ {λ}) = 0$ . Fix $t$ and put $g_{t} (λ^{'}) : = e^{i t λ^{'}} - e^{i t λ}$ , a bounded Borel function vanishing $E_{u}$ -almost everywhere; by [F2], $∥ U (t) u - e^{i t λ} u ∥^{2} = ∥ Φ_{E} (g_{t}) u ∥^{2} = \int ∣ g_{t} ∣^{2} d E_{u} = 0$ , so $U (t) u = e^{i t λ} u$ and, multiplying by $∥ ψ ∥$ , $U (t) ψ = e^{i t λ} ψ$ . In particular $U (t) ψ$ is a multiple of $ψ$ for every $t$ , which is (i). The reality of $λ$ is [F6] applied to the unit vector $u$ .

1.2givenF1F5algebra

[The invariant span] Assume (i). Multiplication by the nonzero scalar $∥ ψ ∥^{- 1}$ shows that $u : = ψ / ∥ ψ ∥$ satisfies $U (t) u = c (t) u$ with $c (t) \in C$ for every $t$ ; since $U (t)$ is unitary and $∥ u ∥ = 1$ , one has $∣ c (t) ∣ = 1$ . By [F5], $K : = C u$ is a closed subspace of $H$ and, with the restricted inner product, a complex Hilbert space; by (i) each $U (t)$ maps $K$ into $K$ , and because $U (t)^{- 1} = U (- t)$ also maps $K$ into $K$ , the restriction $W (t) : = U (t) ∣_{K}$ is a bijection of $K$ onto itself.

2.1givenstep 1.2F1F5

[The restricted group] The maps $W (t)$ are unitary on $K$ : they are norm preserving on $K$ because $U (t)$ is, and they are onto $K$ by step 1.2. The group law $W (t + s) = W (t) W (s)$ and the continuity of $t \mapsto W (t) x$ for each $x \in K$ are inherited from $U$ . Hence $W$ is a strongly continuous one-parameter unitary group on the Hilbert space $K$ .

3.1givenstep 2.1F5F6

[Generator on the span] By [F5] applied to $W$ on $K$ , there is a self-adjoint operator $T_{K}$ on $K$ with $W (t) = e^{i t T_{K}}$ , whose domain $D (T_{K})$ consists exactly of the vectors $x \in K$ for which $\frac{1}{s} (W (s) x - x)$ converges, the limit then being $i T_{K} x$ . A self-adjoint operator is densely defined (Symmetric, self-adjoint and essentially self-adjoint operators), so $D (T_{K})$ is a dense subspace of the one-dimensional space $K$ ; hence $D (T_{K}) = K$ and $T_{K} u = λ u$ for some $λ \in C$ . Since $T_{K}$ is symmetric on $K$ and $u \neq 0$ , [F6] gives $λ \in R$ .

4.1givenstep 3.1F1F5algebra

[Transfer to the ambient space] By step 3.1 and [F5], the difference quotient $\frac{1}{s} (W (s) u - u) = \frac{1}{s} (U (s) u - u)$ converges in $K$ , hence in $H$ , to $i T_{K} u = i λ u$ . By [F1], the quotient of the ambient flow converges exactly for vectors of $D (T)$ , with limit $i T$ applied to the vector; therefore $u \in D (T)$ and $i T u = i λ u$ , that is, $T u = λ u$ with $λ \in R$ . Multiplying by $∥ ψ ∥$ gives $T ψ = λ ψ$ and $ψ \in D (T)$ , which is (ii).

5.1step 1.1step 1.2step 2.1step 3.1step 4.1∎

Steps 1.1, 1.2, 2.1, 3.1 and 4.1 prove the equivalence and the additional display of the statement.

Remarks

Normalization is bookkeeping. The proof normalizes $ψ$ only to use the unit vector in the inner-product computations of [F6] and step 1.1; the statement itself does not require $∥ ψ ∥ = 1$ .
No converse beyond rays. Clause (i) concerns the invariance of the ray; it does not claim that $U (t) ψ = ψ$ for an eigenvector, because the eigenvalue phase $e^{i t λ}$ is present. The statement that the two vectors lie in one ray, hence determine the same pure preparation, is the content of the phase-invariance of states, not an equality of vectors.

Stationary rays of a self-adjoint flow are exactly the eigenrays

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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