LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Closability of smooth magnetic forms with locally integrable potentials

Statement

Assume Countable Choice and Dependent Choice. Let $d, m, ℏ, A, V, v_{0}$ and $q_{A, V}$ be the magnetic form on the test core, with $m, ℏ > 0$ , $A \in C^{\infty} (R^{d}; R^{d})$ , $V \in L_{l o c}^{1}$ real-valued and $V \geq - v_{0}$ almost everywhere. Then $q_{A, V}$ is closable on $C_{c}^{\infty} (R^{d}; C)$ in the shifted form norm $∥ u ∥_{q, v_{0}, ε_{0}}^{2} = q_{A, V} [u] + (v_{0} + ε_{0}) ∥ u ∥_{2}^{2}$ for every $ε_{0} > 0$ . Its closure is a densely defined closed Hermitian form bounded below by $- v_{0}$ .

Statement references

Magnetic quadratic-form data

The Axiom of Countable Choice ( ${A C}_{ω}$ )

The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain

Facts & Assumptions

Given: Countable Choice, Dependent Choice, $d \geq 1$ , $m, ℏ > 0$ , real $q$ , smooth real $A$ , real $V \in L_{l o c}^{1}$ with $V \geq - v_{0}$ , and the core form $q_{A, V}$ .

[A1]

Countable Choice and Dependent Choice are the assumptions of the Hilbert-space, Fubini, and compact-interval integration-by-parts inputs (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[F1]

The complex $L^{2}$ pairing is a Hilbert inner product, $C_{c}^{\infty}$ is dense in $L^{2}$ , and Cauchy–Schwarz holds. The weak-derivative convention uses this same quotient $L^{2}$ space (Magnetic quadratic-form data, Position and momentum operators on Euclidean L2, Cauchy-Schwarz inequality for $L^{2}$ ).

[F2]

Each $Π_{j}^{A}$ maps the core to itself and is symmetric there. On a compact rectangle this follows from one-variable integration by parts on each coordinate slice, Fubini, and the reality of $A_{j}$ (Integration by parts for absolutely continuous functions, Fubini's theorem for L^1 functions on a sigma-finite product).

[F3]

If $W = V + v_{0}$ , then $W \geq 0$ , $W \in L_{l o c}^{1}$ , and $W^{1 / 2} ϕ \in L^{2}$ for every $ϕ \in C_{c}^{\infty}$ : on a compact support, $∣ ϕ ∣$ is bounded and $\int W ∣ ϕ ∣^{2} < \infty$ (Magnetic quadratic-form data).

Proof

1.1givenF1F3algebra

Fix $ε_{0} > 0$ and put $W = V + v_{0}$ . For every $u \in C_{c}^{\infty}$ , $∥ u ∥_{q, v_{0}, ε_{0}}^{2} = \frac{1}{2 m} \sum_{j = 1}^{d} ∥ Π_{j}^{A} u ∥_{2}^{2} + ∥ W^{1 / 2} u ∥_{2}^{2} + ε_{0} ∥ u ∥_{2}^{2} .$ This identity shows that if $(u_{n})$ is Cauchy in the shifted form norm, then each sequence $(Π_{j}^{A} u_{n})$ and $(W^{1 / 2} u_{n})$ is Cauchy in $L^{2}$ .

2.1givenA1F1F2F3step 1.1

Suppose additionally that $∥ u_{n} ∥_{2} \to 0$ . Let $w_{j}$ be the $L^{2}$ limit of $Π_{j}^{A} u_{n}$ . For any $ϕ \in C_{c}^{\infty}$ , symmetry [F2] and Cauchy–Schwarz give $⟨ w_{j}, ϕ ⟩ = \lim_{n} ⟨ Π_{j}^{A} u_{n}, ϕ ⟩ = \lim_{n} ⟨ u_{n}, Π_{j}^{A} ϕ ⟩ = 0.$ Density in [F1] implies $w_{j} = 0$ . If $w_{0}$ is the $L^{2}$ limit of $W^{1 / 2} u_{n}$ , then [F3] and Cauchy–Schwarz similarly give $⟨ w_{0}, ϕ ⟩ = \lim_{n} ⟨ u_{n}, W^{1 / 2} ϕ ⟩ = 0$ for every test function $ϕ$ , so density gives $w_{0} = 0$ .

3.1A1F1step 1.1step 2.1algebra∎

Every term on the right side of the norm identity in step 1.1 therefore tends to zero, including $ε_{0} ∥ u_{n} ∥_{2}^{2}$ . Thus $∥ u_{n} ∥_{q, v_{0}, ε_{0}} \to 0$ , which is exactly the closability criterion in Magnetic quadratic-form data. The completion is densely embedded in $L^{2}$ because it contains the dense core, and the criterion makes that embedding injective; its form is closed by construction and retains the lower bound $- v_{0}$ .

Closability of smooth magnetic forms with locally integrable potentials

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources