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Approximation of Q-P graph-domain vectors by product-domain vectors

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Fix integers $d \geq 1$ , $r \geq 1$ and $ℏ > 0$ , and let $H^{(r)} = L^{2} (R^{d}; C^{r})$ with the componentwise position and momentum operators $Q_{j}, P_{j}$ of Componentwise position and momentum operators on the component-valued wave-function space. Let $ψ \in D (Q_{j}) \cap D (P_{j})$ with $ψ \neq 0$ be a class and let $D_{P} : = D (Q_{j} P_{j}) \cap D (P_{j} Q_{j})$ be the product domain characterized in The position-momentum commutator identity on the product domain clause (b). Then there is a sequence $(φ_{n})_{n \geq 1} \subseteq D_{P}$ with $φ_{n} \to \frac{ψ}{∥ ψ ∥}, Q_{j} φ_{n} \to \frac{Q_{j} ψ}{∥ ψ ∥}, P_{j} φ_{n} \to \frac{P_{j} ψ}{∥ ψ ∥}, ∥ φ_{n} ∥ = 1$ in $H^{(r)}$ . In particular, if $ψ$ is a unit vector, then $φ_{n} \to ψ$ , $Q_{j} φ_{n} \to Q_{j} ψ$ and $P_{j} φ_{n} \to P_{j} ψ$ in norm, and consequently $⟨ Q_{j} φ_{n}, φ_{n} ⟩ \to ⟨ Q_{j} ψ, ψ ⟩, ∥ Q_{j} φ_{n} ∥ \to ∥ Q_{j} ψ ∥, ⟨ P_{j} φ_{n}, φ_{n} ⟩ \to ⟨ P_{j} ψ, ψ ⟩, ∥ P_{j} φ_{n} ∥ \to ∥ P_{j} ψ ∥ .$

Facts & Assumptions

Given: The Axiom of Countable Choice and the Axiom of Choice; integers $d, r \geq 1$ ; $ℏ > 0$ ; the operators $Q_{j}, P_{j}$ with their maximal componentwise domains; a class $ψ \in D (Q_{j}) \cap D (P_{j})$ with $ψ \neq 0$ .

[F1]

Componentwise operators and weak Leibniz. $Q_{j} f = x_{j} f$ and $P_{j} f = - i ℏ D_{j} f$ componentwise on their maximal domains, and $D (P_{j}) = {f : ξ_{j} F_{2}^{(r)} f \in L^{2}}$ (Componentwise position and momentum operators on the component-valued wave-function space). For every smooth $η$ and every $u \in L_{l o c}^{1}$ whose weak $j$ -derivative lies in $L_{l o c}^{1}$ , one has $D_{j} (η u) = (D_{j} η) u + η D_{j} u$ in $L_{l o c}^{1}$ (Weak Leibniz rule with a smooth factor).

[F2]

Cutoffs. There is $χ \in C_{c}^{\infty} (R)$ with $0 \leq χ \leq 1$ , $χ = 1$ on $[- 1, 1]$ , $χ = 0$ outside $[- 2, 2]$ ; for $χ_{n} (t) : = χ (t / n)$ and $n \geq 1$ one has $χ_{n} = 1$ on $[- n, n]$ , $χ_{n} = 0$ outside $[- 2 n, 2 n]$ , $∥ χ_{n} ∥_{\infty} \leq 1$ , and $\sup_{t} ∣ t χ_{n}^{'} (t) ∣ = \sup_{s} ∣ s χ^{'} (s) ∣ = : C_{χ} < \infty$ (Explicit compactly supported smooth cutoffs).

[F3]

Dominated convergence. If $g_{n} \to g$ pointwise and $∣ g_{n} ∣ \leq G$ with $\int G < \infty$ , then $\int g_{n} \to \int g$ (Dominated convergence).

[F4]

The product domain. $D_{P} = {f \in H^{(r)} : D_{j} f, x_{j} f, x_{j} D_{j} f \in L^{2} (R^{d}; C^{r})}$ and on $D_{P}$ the product rule $D_{j} (x_{j} f) = f + x_{j} D_{j} f$ holds (The position-momentum commutator identity on the product domain clauses (a) and (b)).

[F5]

Inner-product continuity. The inner product and the norm of $H^{(r)}$ are continuous for norm convergence (Hilbert space, The space $L^{p} (μ)$ as the quotient by null functions).

Proof

technique · cut the vector off in the $j$-th coordinate by a smooth cutoff with bounded logarithmic derivative, verify the product-domain memberships with the weak Leibniz rule, and pass to the limit by dominated convergence before normalizing

Given: The data of the statement and [F1]-[F5].

1.1givenF2

[The truncated vector] For $n \geq 1$ put ${\tilde{φ}}_{n} : = χ_{n} (x_{j}) ψ$ , the componentwise product of the class $ψ$ with the bounded smooth function $χ_{n} (x_{j})$ . Since $∣ χ_{n} ∣ \leq 1$ and $ψ \in L^{2}$ , the class ${\tilde{φ}}_{n} \in H^{(r)}$ satisfies $∥ {\tilde{φ}}_{n} ∥ \leq ∥ ψ ∥$ .

2.1givenstep 1.1F1F2F4algebra

[Membership in the product domain] Since $ψ \in D (P_{j})$ , [F1] with $η = χ_{n}$ gives $D_{j} {\tilde{φ}}_{n} = χ_{n}^{'} ψ + χ_{n} D_{j} ψ$ , a sum of two $L^{2}$ classes because $χ_{n}^{'}$ and $χ_{n}$ are bounded; hence ${\tilde{φ}}_{n} \in D (P_{j})$ . Next $x_{j} D_{j} {\tilde{φ}}_{n} = x_{j} χ_{n}^{'} ψ + x_{j} χ_{n} D_{j} ψ$ : the first summand is bounded in modulus by $C_{χ} ∣ ψ ∣$ pointwise by [F2], and the second by $2 n ∣ D_{j} ψ ∣$ because $∣ x_{j} χ_{n} (x_{j}) ∣ \leq 2 n$ on the support of $χ_{n}$ and $D_{j} ψ \in L^{2}$ since $ψ \in D (P_{j})$ ; hence $x_{j} D_{j} {\tilde{φ}}_{n} \in L^{2}$ . Also $x_{j} {\tilde{φ}}_{n} = x_{j} χ_{n} ψ \in L^{2}$ because $∣ x_{j} χ_{n} (x_{j}) ∣ \leq 2 n$ is bounded. By [F4] the three memberships place ${\tilde{φ}}_{n}$ in $D_{P}$ .

2.2step 1.1F1F2F3algebra

[Convergence of the truncated sequence] As $n \to \infty$ one has $χ_{n} (x_{j}) \to 1$ , $χ_{n} (x_{j})^{'} \to 0$ and $x_{j} χ_{n}^{'} (x_{j}) \to 0$ pointwise, with the uniform bounds $∣ χ_{n} ∣ \leq 1$ , $∣ χ_{n}^{'} ∣ \leq ∥ χ^{'} ∥_{\infty} / n$ and $∣ x_{j} χ_{n}^{'} ∣ \leq C_{χ}$ . Apply [F3] componentwise to the squared moduli: $∣ χ_{n} ψ - ψ ∣^{2} \leq ∣ ψ ∣^{2}$ gives ${\tilde{φ}}_{n} \to ψ$ in $H^{(r)}$ , and $∣ x_{j} (χ_{n} - 1) ψ ∣^{2} \leq ∣ x_{j} ψ ∣^{2}$ gives $Q_{j} {\tilde{φ}}_{n} \to Q_{j} ψ$ . Both majorants are integrable by $ψ \in D (Q_{j})$ . For the momentum, $∥ χ_{n}^{'} ψ ∥_{2} \leq ∥ χ^{'} ∥_{\infty} ∥ ψ ∥_{2} / n \to 0$ , while $∣ (χ_{n} - 1) D_{j} ψ ∣^{2} \leq ∣ D_{j} ψ ∣^{2}$ and pointwise convergence give $χ_{n} D_{j} ψ \to D_{j} ψ$ in $L^{2}$ by [F3]; the last majorant is integrable because $ψ \in D (P_{j})$ . Therefore $Q_{j} {\tilde{φ}}_{n} = x_{j} χ_{n} ψ \to x_{j} ψ = Q_{j} ψ, P_{j} {\tilde{φ}}_{n} = - i ℏ (χ_{n}^{'} ψ + χ_{n} D_{j} ψ) \to - i ℏ D_{j} ψ = P_{j} ψ$ in $H^{(r)}$ . In particular $∥ {\tilde{φ}}_{n} ∥ \to ∥ ψ ∥ > 0$ .

3.1step 2.2F4algebra

[Normalization] Since $∥ {\tilde{φ}}_{n} ∥ \to ∥ ψ ∥ > 0$ , choose $n_{0}$ such that ${\tilde{φ}}_{n} \neq 0$ for every $n \geq n_{0}$ . Reindex this tail by setting $φ_{n} : = {\tilde{φ}}_{n + n_{0} - 1} / ∥ {\tilde{φ}}_{n + n_{0} - 1} ∥$ for every $n \geq 1$ . Since $D_{P}$ is a linear subspace by [F4], $φ_{n} \in D_{P}$ and $∥ φ_{n} ∥ = 1$ . Dividing the convergences of step 2.2 by $∥ {\tilde{φ}}_{n + n_{0} - 1} ∥ \to ∥ ψ ∥$ gives $φ_{n} \to ψ / ∥ ψ ∥$ , $Q_{j} φ_{n} \to Q_{j} ψ / ∥ ψ ∥$ and $P_{j} φ_{n} \to P_{j} ψ / ∥ ψ ∥$ . If $ψ$ is a unit vector, these are the convergences $φ_{n} \to ψ$ , $Q_{j} φ_{n} \to Q_{j} ψ$ and $P_{j} φ_{n} \to P_{j} ψ$ .

4.1step 3.1F5algebra∎

[Means and second moments] Assume $ψ$ is a unit vector. By [F5], $⟨ Q_{j} φ_{n}, φ_{n} ⟩ \to ⟨ Q_{j} ψ, ψ ⟩$ and $∥ Q_{j} φ_{n} ∥ = ∥ Q_{j} φ_{n} - 0 ∥ \to ∥ Q_{j} ψ ∥$ from the convergence $Q_{j} φ_{n} \to Q_{j} ψ$ ; the same two limits hold with $Q_{j}$ replaced by $P_{j}$ . This completes the statement.

Approximation of Q-P graph-domain vectors by product-domain vectors

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources