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Unitary groups converge under strong resolvent convergence

Statement

Assume the Axiom of Choice. Let $A_{n}, A$ be self-adjoint operators on a complex Hilbert space $H$ with $A_{n} \to A$ in the strong resolvent sense. Then $e^{i t A_{n}} x ⟶ e^{i t A} x for every t \in R and every x \in H .$ If, in addition, $A$ and all $A_{n}$ are bounded below by a common real constant $γ$ (meaning $⟨ S x, x ⟩ \geq γ ∥ x ∥^{2}$ for $x \in D (S)$ and $S = A, A_{n}$ ), then $e^{- t A_{n}} \to e^{- t A}$ strongly for every $t \geq 0$ .

Facts & Assumptions

[A1]

For a self-adjoint $S$ with spectral PVM, $e^{i t S}$ is the Borel calculus of $λ \mapsto e^{i t λ}$ and is a strongly continuous unitary group, with infinitesimal generator $i S$ (A self-adjoint operator generates a strongly continuous unitary group, Infinitesimal generator of a unitary group, Strongly continuous one-parameter unitary group).

[A2]

Under AC, strong resolvent convergence gives $g (A_{n}) x \to g (A) x$ for every bounded continuous $g : R \to C$ and every $x$ (Continuous functional calculus under resolvent convergence, The Axiom of Choice).

[A3]

On nonzero $H$ , each self-adjoint $S$ has a spectral PVM $E^{S}$ with $D (S) = {x : \int λ^{2} d E_{x}^{S} < \infty}$ and $S = \int λ d E^{S}$ . For domain vectors, $⟨ S x, x ⟩ = \int λ d E_{x}^{S}$ , using the quadratic pairing identity and the reality of $λ$ in the first-variable-linear convention. Functions agreeing off a measurable $E^{S}$ -null set have the same integral operator and domain; the zero-space calculus is defined directly (Spectral theorem for unbounded self-adjoint operators (PVM form), The unbounded PVM integral is densely defined, closed and normal, Integral of a measurable function against a projection-valued measure).

[A4]

PVM projections satisfy $E (B) E (C) = E (B \cap C)$ and $E (R) = I$ ; the scalar measures are positive of mass $∥ x ∥^{2}$ , and scalar monotone convergence holds (Projection valued measure, Scalar and complex measures from a pvm, Monotone convergence for the integral).

Proof

technique · direct

Given: AC and the self-adjoint operators with strong resolvent convergence in the statement.

1.1A1A2A3given

If $H = {0}$ , all operators in either conclusion are its unique operator, so both conclusions hold. Otherwise the spectral PVMs exist by [A3]. Fix any $t \in R$ , including negative times. The function $λ \mapsto e^{i t λ}$ is continuous and has absolute value one everywhere. Thus [A2] gives $e^{i t A_{n}} x \to e^{i t A} x$ for every $x$ ; [A1] identifies these as the stated unitary groups. At $t = 0$ each operator is $I$ .

1.2A3A4given

For the second claim suppose $S$ is one of $A, A_{n}$ and satisfies the common lower bound. Let $B_{m} = [- m, γ - 1 / m]$ for integers $m \geq 1$ , with an empty interval interpreted as empty. If $E^{S} (B_{m}) \neq 0$ , choose $x = E^{S} (B_{m}) y \neq 0$ for some $y$ . The projection identities give $E^{S} (R ∖ B_{m}) x = 0$ , so the scalar measure of $x$ is supported on $B_{m}$ . As $B_{m}$ is bounded, [A3] gives $x \in D (S)$ and $⟨ S x, x ⟩ = \int λ d E_{x}^{S} \leq (γ - 1 / m) ∥ x ∥^{2} < γ ∥ x ∥^{2}$ , a contradiction. Hence each $E^{S} (B_{m}) = 0$ . The sets $B_{m}$ increase to $(- \infty, γ)$ ; monotone convergence gives $E_{x}^{S} ((- \infty, γ)) = 0$ for every $x$ . Since $∥ E^{S} (B) x ∥^{2} = ⟨ E^{S} (B) x, x ⟩$ , the projection $E^{S} ((- \infty, γ))$ itself is zero.

2.1A2A3A4step 1.2

Now fix $t \geq 0$ and define $g_{t} (λ) = e^{- t \max (λ, γ)}$ . It is continuous and bounded by $e^{- t γ}$ , and it agrees with $e^{- t λ}$ on $[γ, \infty)$ . Step 1.2 and null-set invariance in [A3] give equality of the integral operators $g_{t} (S) = e^{- t S}$ , including domains, for $S = A, A_{n}$ . In particular each exponential here has full domain and is bounded: its defining squared integral is at most $e^{- 2 t γ} ∥ x ∥^{2}$ , and its quadratic norm identity gives the same operator bound. Applying [A2] to $g_{t}$ proves $e^{- t A_{n}} x \to e^{- t A} x$ for every $x$ .

3.1A1A2A3step 1.1step 2.1∎

The first conclusion holds for every real time by step 1.1, and the second for every nonnegative time by step 2.1. At time zero both reduce to the identity. The lower bound is assumed for the limit and every approximant; no preservation-of-lower-bound theorem is assumed. AC supplies the spectral and calculus hypotheses, including their Countable Choice assumptions.

Depends on

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Sources

Gerald Teschl, Mathematical Methods in Quantum Mechanics, second edition (standard reference, not scraped)
Roland Schnaubelt, Evolution Equations (lecture notes) (standard reference, not scraped)