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Driven oscillator common domain

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $m, ω, ℏ > 0$ , let $ℓ : = \sqrt{ℏ / (m ω)}$ , let $H_{o s c} = \frac{ℏ ω}{2} D A D^{- 1}$ be the oscillator of Oscillator scaling and exact realization with orthonormal eigenbasis $(ψ_{n})_{n \geq 0}$ and ladder operators $a_{ψ}, a_{ψ}^{†}, N_{ψ}$ of Closed ladder domains transported to $L^{2} (R, d x)$ . Work with $⟨ u, v ⟩ = \int u \overline{v}$ , linear in its first argument, and define $N_{ψ}^{1 / 2}$ by the coefficient action $\sqrt{n} c_{n}$ on $D (N_{ψ}^{1 / 2}) = {ψ : \sum_{n} n ∣ c_{n} ∣^{2} < \infty}$ , where $c_{n} = ⟨ ψ, ψ_{n} ⟩$ . Use the negative-sign $2 π$ Fourier convention $\hat{u} (ξ) = \int e^{- 2 π i x ξ} u (x) d x$ on integrable inputs and its Plancherel extension $F_{2}$ . Let $Q = M_{x}$ , $P = F_{2}^{*} M_{2 π ℏ ξ} F_{2}$ be the maximal position and momentum operators (their self-adjointness is proved below), with $D (Q) = {ψ : x ψ \in L^{2}}$ and $D (P) = {ψ : ξ F_{2} ψ \in L^{2}}$ , with $H_{o s c} = ℏ ω (N_{ψ} + \frac{1}{2} I)$ and the couplings $Q = \frac{ℓ}{\sqrt{2}} (a_{ψ} + a_{ψ}^{†}), P = \frac{ℏ}{i ℓ \sqrt{2}} (a_{ψ} - a_{ψ}^{†})$ on the common domain $D (N_{ψ}^{1 / 2})$ . Let $f : [a, b] \to R$ be of class $C^{1}$ . Put $Y : = D (H_{o s c}) = D (N_{ψ})$ with graph norm $∥ ψ ∥_{Y}^{2} : = ∥ ψ ∥^{2} + ∥ H_{o s c} ψ ∥^{2}$ . Then:

(a) Ladder bounds. For every $ψ \in D (N_{ψ}^{1 / 2})$ , $∥ Q ψ ∥ \leq ℓ (2 ∥ N_{ψ}^{1 / 2} ψ ∥^{2} + ∥ ψ ∥^{2})^{1 / 2}, ∥ P ψ ∥ \leq \frac{ℏ}{ℓ} (2 ∥ N_{ψ}^{1 / 2} ψ ∥^{2} + ∥ ψ ∥^{2})^{1 / 2} .$

(b) Infinitesimal bound. For every $η > 0$ there is $b_{η} < \infty$ with $∥ Q ψ ∥ \leq η ∥ H_{o s c} ψ ∥ + b_{η} ∥ ψ ∥$ for $ψ \in Y = D (H_{o s c})$ ; hence $- f (t) Q$ is $H_{o s c}$ -bounded with relative bound $0$ for every fixed $t$ (Relative boundedness with respect to an operator).

(c) Common domain. For every $t \in [a, b]$ the operator $H (t) : = H_{o s c} - f (t) Q$ is self-adjoint on the single domain $Y$ (independent of $t$ ); moreover $t \mapsto H (t)$ belongs to $C^{1} ([a, b]; B (Y, L^{2}))$ with derivative $- f^{'} (t) Q$ , so the family has a common domain and satisfies the regularity hypothesis of Common-domain nonautonomous evolution.

Facts & Assumptions

Given: CC; $m, ω, ℏ > 0$ , $ℓ = \sqrt{ℏ / (m ω)}$ ; the scaled Hermite basis $(ψ_{n})$ and its diagonal oscillator $A : = H_{o s c}$ ; $f \in C^{1} ([a, b]; R)$ ; $Y = D (A)$ with its stated graph norm. Write $c : = ℏ ω > 0$ .

[F1]

Oscillator scaling and exact realization assumes CC and supplies the exact diagonal domain $Y = {ψ : \sum_{n} c^{2} (n + 1 / 2)^{2} ∣ ⟨ ψ, ψ_{n} ⟩ ∣^{2} < \infty}$ , self-adjointness of $A$ , its closedness and graph-core finite sums. This is also $D (N_{ψ})$ , with $A = c (N_{ψ} + I / 2)$ ; the equality follows directly from the displayed coefficient actions. Complex $L^{2}$ is complete under CC by Complex completeness, density, and inner product: the consumer interface, so its graph domain is a Hilbert space because its graph is closed in $L^{2} \oplus L^{2}$ .

[F2]

Closed ladder domains supplies the exact shifted coefficient actions under CC. Define $N_{ψ}^{1 / 2}$ directly by $(N_{ψ}^{1 / 2} ψ)_{n} = \sqrt{n} c_{n}$ on ${ψ : \sum_{n} n ∣ c_{n} ∣^{2} < \infty}$ , where $c_{n} = ⟨ ψ, ψ_{n} ⟩$ . For this domain, $∥ a_{ψ} ψ ∥^{2} = \sum_{n} n ∣ c_{n} ∣^{2} = : A_{ψ}$ and $∥ a_{ψ}^{†} ψ ∥^{2} = A_{ψ} + ∥ ψ ∥^{2}$ . Finite coefficient truncations converge in both ladder graph norms.

[F3]

Hermite recurrence gives $\sqrt{2} y g_{n} = \sqrt{n} g_{n - 1} + \sqrt{n + 1} g_{n + 1}$ and $\sqrt{2} g_{n}^{'} = \sqrt{n} g_{n - 1} - \sqrt{n + 1} g_{n + 1}$ . Scaling by the actual $D$ gives the finite-basis formulas for $Q$ and $P$ used below. Plancherel theorem and Distributional derivatives are polynomial Fourier multipliers assume CC and identify $P ψ_{n} = - i ℏ ψ_{n}^{'}$ via its exact Fourier multiplier. Every $ψ_{n}$ and its derivative belong to $L^{2}$ .

[F4]

The adjoint-domain definition Adjoint of a densely defined operator and closedness of adjoints The adjoint is well defined, closed, and reverses inclusions are CC interfaces. A vector $y$ belongs to $D (T^{*})$ precisely when there is $w$ with $⟨ T x, y ⟩ = ⟨ x, w ⟩$ for all $x \in D (T)$ ; the inner product is linear in its first argument. Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors bounds all Hilbert pairings. Dominated convergence justifies the explicit multiplier truncations below. No Kato-Rellich theorem or general AC spectral supplier is used.

Proof

1.1F3F4givenconstructalgebra

[Concrete maximal multipliers are self-adjoint under CC] For any real measurable finite-valued $v$ on $R$ , define $T_{v} h = v h$ on ${h \in L^{2} : v h \in L^{2}}$ . For $h \in L^{2}$ , $h_{n} = 1_{{∣ v ∣ \leq n}} h$ belongs to this domain and converges to $h$ by dominated convergence against $∣ h ∣^{2}$ , so the domain is dense. The real multiplier gives symmetry. If $y \in D (T_{v}^{*})$ and $w = T_{v}^{*} y$ , test the adjoint identity against every $x \in L^{2}$ supported on $E_{n} = {∣ v ∣ \leq n}$ ; each such $x$ is in $D (T_{v})$ , and $⟨ x, v y - w ⟩ = 0$ . Taking $x = 1_{E_{n}} (v y - w) \in L^{2}$ gives $v y = w$ a.e. on $E_{n}$ . Their union is all $R$ , so $v y = w \in L^{2}$ , whence $y \in D (T_{v})$ and $T_{v}^{*} y = T_{v} y$ . The converse follows by symmetry, proving self-adjointness and thus closedness by [F4]. Apply this to $v (x) = x$ and $v (ξ) = 2 π ℏ ξ$ . Writing $U = F_{2}$ and $T = T_{2 π ℏ ξ}$ , the adjoint identity for $U^{*} T U$ tested on $u = U^{*} v$ , $v \in D (T)$ , says $y \in D ((U^{*} T U)^{*})$ exactly when $U y \in D (T^{*}) = D (T)$ , and then $(U^{*} T U)^{*} y = U^{*} T U y$ . Thus $Q = M_{x}$ and $P = F_{2}^{*} T_{2 π ℏ ξ} F_{2}$ are self-adjoint and closed on exactly their maximal displayed domains. This reconstructs the needed concrete operators without invoking the AC-bearing general composite carrier.

1.2F1givenconstructalgebra

[Actual diagonal nonreal inverses] For $R > 0$ and $z \in {i R, - i R}$ define $R_{z} h : = \sum_{n \geq 0} \frac{⟨ h, ψ_{n} ⟩}{c (n + 1 / 2) - z} ψ_{n} .$ The real positive diagonal values $λ_{n} = c (n + 1 / 2)$ give $∣ λ_{n} - z ∣ \geq R$ and $∣ λ_{n} / (λ_{n} - z) ∣ \leq 1$ . Hence the series converges, $∥ R_{z} ∥ \leq 1 / R$ , $R_{z} h \in Y$ and $∥ A R_{z} ∥ \leq 1$ . Direct coefficient multiplication verifies $(A - z) R_{z} = I$ on $L^{2}$ and $R_{z} (A - z) = I$ on $Y$ . In particular $R_{z} : L^{2} \to Y$ is bounded for the fixed graph norm, with $∥ R_{z} h ∥_{Y}^{2} \leq (1 + R^{- 2}) ∥ h ∥^{2}$ . These are concrete countable diagonal formulas, without a spectral theorem.

2.1step 1.1F2F3algebra

[The complete common half-number domain] By [F3], for every finite expansion the coupling identities in the Statement hold. Let $ψ \in D (N_{ψ}^{1 / 2})$ and $ψ_{M} = \sum_{n \leq M} c_{n} ψ_{n}$ . By [F2], $ψ_{M} \to ψ$ and both ladder images converge to their stated series. Thus $Q ψ_{M}$ and $P ψ_{M}$ converge to the respective ladder combinations. Closedness from step 1.1 places $ψ$ in both $D (Q)$ and $D (P)$ and gives the coupling identities on all $D (N_{ψ}^{1 / 2})$ . For $A_{ψ} = ∥ N_{ψ}^{1 / 2} ψ ∥^{2}$ and $B_{ψ} = ∥ ψ ∥^{2}$ , the triangle inequality and $(x + y)^{2} \leq 2 (x^{2} + y^{2})$ give $∥ Q ψ ∥^{2} \leq \frac{ℓ^{2}}{2} (\sqrt{A_{ψ}} + \sqrt{A_{ψ} + B_{ψ}})^{2} \leq ℓ^{2} (2 A_{ψ} + B_{ψ}), ∥ P ψ ∥^{2} \leq \frac{ℏ^{2}}{ℓ^{2}} (2 A_{ψ} + B_{ψ}) .$ Taking nonnegative square roots proves the corrected full-domain (a). No assumption restricts $ψ$ to coherent states or finite expansions.

3.1step 2.1F1F2algebra

[Explicit infinitesimal relative bound] For $ψ \in Y$ , coefficient Cauchy-Schwarz gives $A_{ψ} = \sum_{n} n ∣ c_{n} ∣^{2} \leq ∥ N_{ψ} ψ ∥ ∥ ψ ∥$ . The diagonal formula gives $∥ N_{ψ} ψ ∥ \leq c^{- 1} ∥ A ψ ∥$ because $n \leq n + 1 / 2$ coefficientwise. Set $X = ∥ N_{ψ} ψ ∥$ and $B = ∥ ψ ∥$ . Step 2.1 implies $∥ Q ψ ∥ \leq ℓ \sqrt{2 X B + B^{2}} \leq ℓ \sqrt{2 X B} + ℓ B$ . For any $δ > 0$ , the nonnegative square $(\sqrt{δ X} - \sqrt{B / (2 δ)})^{2}$ gives $\sqrt{2 X B} \leq δ X + B / (2 δ)$ . Given the Statement's $η > 0$ , take $δ = η c / ℓ$ to obtain $∥ Q ψ ∥ \leq η ∥ A ψ ∥ + b_{η} ∥ ψ ∥, b_{η} : = ℓ + \frac{ℓ^{2}}{2 η c} < \infty .$ Multiplication by the fixed finite real number $∣ f (t) ∣$ makes the relative $A$ -bound of $- f (t) Q$ zero; when $f (t) = 0$ the perturbation vanishes. This proves all of (b).

4.1step 3.1step 1.2F1constructalgebra

[Two uniform Neumann inverses on the original domain] Set $F = \sup_{t \in [a, b]} ∣ f (t) ∣ < \infty$ , $η_{0} = 1 / (4 \max (1, F))$ , $β = F b_{η_{0}}$ and choose $R > 4 β + 1$ . For $B_{t} = - f (t) Q$ restricted to $Y$ , step 3.1 gives $∥ B_{t} u ∥ \leq (1 / 4) ∥ A u ∥ + β ∥ u ∥$ uniformly in $t$ . Therefore $B_{t} R_{z}$ is a bounded operator on $L^{2}$ with norm at most $1 / 4 + β / R < 1 / 2$ for both $z = \pm i R$ , by step 1.2. Its Neumann series $L_{t, z} = \sum_{j \geq 0} (- B_{t} R_{z})^{j}$ converges in operator norm, since the tail norm is bounded by the geometric tail $\sum_{j > J} 2^{- j}$ ; completeness of $B (L^{2})$ follows from pointwise norm limits and the uniform bound. Multiplying finite partial sums and taking their norm limit gives $L_{t, z} (I + B_{t} R_{z}) = (I + B_{t} R_{z}) L_{t, z} = I$ . On $Y$ the exact factorization is $H (t) - z = (I + B_{t} R_{z}) (A - z)$ . Consequently $S_{t, z} : = R_{z} L_{t, z} : L^{2} \to Y$ is its two-sided inverse, including $S_{t, z} (H (t) - z) u = u$ for every $u \in Y$ . Thus both $H (t) \pm i R$ are onto and injective on the original domain; no larger domain is introduced by the inverse construction.

5.1step 1.1step 4.1F1F4algebra

[Exact adjoint domain] Since $f (t)$ is real and $Y \subseteq D (Q)$ , steps 1.1 and [F1] imply that $H (t) = A + B_{t}$ on the dense domain $Y$ is symmetric. Hence $Y \subseteq D (H (t)^{*})$ and the adjoint agrees with $H (t)$ there. Let $y \in D (H (t)^{*})$ and put $g = (H (t)^{*} - i R) y \in L^{2}$ . Step 4.1 supplies $x = S_{t, i R} g \in Y$ with $(H (t) - i R) x = g$ . Then $w = y - x$ lies in $D (H (t)^{*})$ and $(H (t)^{*} - i R) w = 0$ . For every $u \in Y$ , the first-variable-linear adjoint convention gives $⟨ (H (t) + i R) u, w ⟩ = ⟨ u, H (t)^{*} w ⟩ + i R ⟨ u, w ⟩ = 0$ , since $H (t)^{*} w = i R w$ makes the first term $- i R ⟨ u, w ⟩$ . The range of $H (t) + i R$ is all $L^{2}$ by step 4.1, so $w$ is orthogonal to all of $L^{2}$ and $w = 0$ . Thus $y = x \in Y$ and $H (t)^{*} y = H (t) y$ . The reverse inclusion was already shown, proving exact self-adjointness with $D (H (t)^{*}) = D (H (t)) = Y = D (N_{ψ})$ for every $t$ . This supplies the missing conclusion directly under CC.

6.1step 3.1step 4.1step 5.1F1algebra

[Common graph regularity] Step 3.1 with $η = 1$ shows $∥ Q ψ ∥ \leq (1 + b_{1}) ∥ ψ ∥_{Y}$ , so $Q : Y \to L^{2}$ is bounded; also $A : Y \to L^{2}$ is bounded with norm at most one. Therefore $H (t) = A - f (t) Q \in B (Y, L^{2})$ and $∥ H (t) - H (s) ∥_{Y \to L^{2}} \leq ∣ f (t) - f (s) ∣ ∥ Q ∥_{Y \to L^{2}}$ . For interior $t$ , and with the corresponding one-sided endpoint limits, ${∥ \frac{H (t + h) - H (t)}{h} + f^{'} (t) Q ∥}_{Y \to L^{2}} \leq ∣ \frac{f (t + h) - f (t)}{h} - f^{'} (t) ∣ ∥ Q ∥_{Y \to L^{2}} ⟶ 0.$ Continuity of $f^{'}$ makes the derivative norm-continuous, proving $C^{1} ([a, b]; B (Y, L^{2}))$ . The fixed graph space $Y$ is dense and Hilbert by [F1]. If needed, step 4.1's bound also gives uniformly equivalent moving graph norms: $∥ H (t) u ∥ \leq (5 / 4) ∥ A u ∥ + β ∥ u ∥$ and $(3 / 4) ∥ A u ∥ \leq ∥ H (t) u ∥ + β ∥ u ∥$ . Thus all common-domain and regularity hypotheses claimed in (c), including those of Common-domain nonautonomous evolution after an affine change of the time interval, are met. No propagator formula or extra Weyl-domain hypothesis is assumed.

7.1step 2.1step 3.1step 5.1step 6.1∎

[Conclusion] Step 2.1 proves the corrected (a), step 3.1 proves (b), and steps 5.1 and 6.1 prove the entire common-domain and norm- $C^{1}$ assertion (c) on $D (N_{ψ})$ .

Driven oscillator common domain

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources