TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Fixed-N Coulomb self-adjointness

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $N \geq 1$ and $r \geq 1$ , let $m_{1}, \dots, m_{N} > 0$ , $ℏ > 0$ , let $a_{1}, \dots, a_{M} \in R^{3}$ be fixed points, and let $z_{i m}, z_{i j} \in R$ be real coefficients ( $1 \leq i \leq N$ , $1 \leq m \leq M$ , $1 \leq i < j \leq N$ ). Let $K$ be the free kinetic operator of Bounded fixed-N operator, $D (K) = W^{2, 2} (R^{3 N}; C)$ with equivalent graph norm, and define the singular multiplier $V (X) : = \sum_{i = 1}^{N} \sum_{m = 1}^{M} \frac{z_{i m}}{∣ x_{i} - a_{m} ∣} + \sum_{1 \leq i < j \leq N} \frac{z_{i j}}{∣ x_{i} - x_{j} ∣}, X = (x_{1}, \dots, x_{N}) \in R^{3 N},$ as an a.e.-defined real measurable function (The space $L^{p} (μ)$ as the quotient by null functions). Then:

(a) each of the multipliers $u \mapsto u / ∣ x_{i} - a_{m} ∣$ and $u \mapsto u / ∣ x_{i} - x_{j} ∣$ is $K$ -bounded with $K$ -bound zero: for every $η > 0$ there is a finite constant $C_{η}$ with $∥ \frac{u}{∣ x_{i} - a_{m} ∣} ∥_{2} \leq η ∥ K u ∥_{2} + C_{η} ∥ u ∥_{2}, ∥ \frac{u}{∣ x_{i} - x_{j} ∣} ∥_{2} \leq η ∥ K u ∥_{2} + C_{η} ∥ u ∥_{2}$ for all $u \in D (K)$ (Relative boundedness with respect to an operator, Weak derivative of a locally integrable function);

(b) $D (K) \subseteq D (M_{V})$ , and $M_{V}$ is symmetric on $D (K)$ with $K$ -bound zero; consequently $H : = K + M_{V}, D (H) : = D (K) = W^{2, 2} (R^{3 N}; C),$ is self-adjoint and bounded below, and $C_{c}^{\infty} (R^{3 N}; C)$ is a core for $H$ (Kato-Rellich theorem, Symmetric, self-adjoint and essentially self-adjoint operators);

(c) the operator $K \otimes I_{r} + M_{V} \otimes I_{r}$ on $L^{2} (R^{3 N}; C) \otimes C^{r}$ with domain $D (K) \otimes C^{r}$ is self-adjoint and bounded below;

(d) if $m_{1} = \dots = m_{N} = m$ and the coefficients are permutation invariant ( $z_{i m}$ independent of $i$ and $z_{i j}$ independent of the pair), then the transported permutation unitaries of Permutation-restricted self-adjointness reduce $H$ , and the restrictions of $H$ to the symmetric and antisymmetric sectors are self-adjoint on those sectors.

No Hunziker-van Winter-Zhislin threshold statement, no uniform-in- $N$ stability, no thermodynamic limit, no spectral statement beyond self-adjointness and boundedness from below, and no physical interpretation of $H$ , the masses or the coefficients is asserted.

Facts & Assumptions

Given: the integers $N, r \geq 1$ , masses $m_{i} > 0$ , the constant $ℏ > 0$ , fixed points $a_{m}$ , real coefficients $z_{i m}, z_{i j}$ , the free kinetic operator $K$ with domain $D (K) = W^{2, 2} (R^{3 N})$ , the partial Laplacians $Δ_{i}$ and the operators $K_{i} : = - ℏ^{2} Δ_{i} / (8 π^{2} m_{i})$ on the $i$ -th block, and the multiplier $V$ defined above.

[F1]

$K = \sum_{i} K_{i}$ is self-adjoint, positive, with $D (K) = W^{2, 2} (R^{3 N}; C)$ and graph norm equivalent to the $W^{2, 2}$ norm; for every $u \in D (K)$ : $∥ \nabla_{i} u ∥_{2}^{2} = (8 π^{2} m_{i} / ℏ^{2}) ⟨ K_{i} u, u ⟩ \leq (8 π^{2} m_{i} / ℏ^{2}) ⟨ K u, u ⟩ \leq (8 π^{2} m_{i} / ℏ^{2}) ∥ K u ∥_{2} ∥ u ∥_{2}$ by Cauchy-Schwarz in $L^{2}$ , and $∥ Δ_{i} u ∥_{2} \leq (8 π^{2} m_{i} / ℏ^{2}) ∥ K u ∥_{2}$ (Bounded fixed-N operator, Integer-order W^{k,2} and H^k agree with equivalent norms, Plancherel theorem, Integer-order Sobolev spaces and their norms, Cauchy-Schwarz inequality for $L^{2}$ ).

[F2]

Three-dimensional Hardy inequality: for $v \in W^{1, 2} (R^{3}; C)$ , $∥ v / ∣ \cdot ∣ ∥_{2} \leq 2 ∥ \nabla v ∥_{2}$ (Three-dimensional Hardy bound (c)).

[F3]

$C_{c}^{\infty} (R^{3 N}; C)$ is dense in $W^{1, 2} (R^{3 N}; C)$ and in $W^{2, 2}$ ; under Countable Choice $L^{2}$ convergence supplies an a.e.-convergent subsequence, and Fatou bounds the integral of the nonnegative pointwise limit by the liminf of integrals (Compactly supported smooth functions are dense in W^{k,p}(R^n), Complex Lp completeness and almost-everywhere subsequences, Fatou's lemma, The space $L^{p} (μ)$ as the quotient by null functions). Tonelli permits the nonnegative slice integrations over the sigma-finite Lebesgue product spaces (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F4]

Translations preserve Lebesgue measure, and the orthogonal map $(x_{i}, x_{j}) \mapsto (y, z) = ((x_{i} - x_{j}) / \sqrt{2}, (x_{i} + x_{j}) / \sqrt{2})$ preserves Lebesgue measure, hence $L^{2}$ norms and integrals; under it, $\partial_{x_{i}} = 2^{- 1 / 2} (\partial_{z} + \partial_{y})$ , $\partial_{x_{j}} = 2^{- 1 / 2} (\partial_{z} - \partial_{y})$ and $∣ \nabla_{y} w ∣^{2} + ∣ \nabla_{z} w ∣^{2} = ∣ \nabla_{i} w ∣^{2} + ∣ \nabla_{j} w ∣^{2}$ (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation, Lebesgue measure on $R^{n}$ is invariant under every orthogonal linear map, Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}), Weak derivative of a locally integrable function).

[F5]

Kato-Rellich: if $A$ is self-adjoint, $B$ symmetric with $D (A) \subseteq D (B)$ and $A$ -bound less than $1$ , then $A + B$ on $D (A)$ is self-adjoint and bounded below when $A$ is; a finite sum of $A$ -bound-zero symmetric operators has $A$ -bound zero (Kato-Rellich theorem, Relative boundedness with respect to an operator).

Proof

1.1F2F3F4algebra

[Global Hardy inequalities] Let $u \in C_{c}^{\infty} (R^{3 N})$ and fix a block index $i$ and a center $a \in R^{3}$ . For each fixed value of the remaining coordinates the slice $x_{i} \mapsto u (\dots, x_{i}, \dots)$ is a compactly supported smooth function on $R^{3}$ ; [F2] gives $\int_{R^{3}} ∣ u ∣^{2} / ∣ x_{i} - a ∣^{2} d x_{i} \leq 4 \int_{R^{3}} ∣ \nabla_{i} u ∣^{2} d x_{i}$ for that slice, and integrating over the remaining coordinates by Tonelli [F3] with [F4] (translation to the center $a$ ) yields $\int ∣ u ∣^{2} / ∣ x_{i} - a ∣^{2} \leq 4 \int ∣ \nabla_{i} u ∣^{2}$ . For a pair replace $(x_{i}, x_{j})$ by the orthogonal coordinates $(y, z)$ of [F4]: then $∣ x_{i} - x_{j} ∣^{2} = 2 ∣ y ∣^{2}$ and, applying the same slice argument in the $y$ -variable to the transformed function, $\int ∣ u ∣^{2} / ∣ x_{i} - x_{j} ∣^{2} = \frac{1}{2} \int ∣ u^{'} ∣^{2} / ∣ y ∣^{2} \leq 2 \int ∣ \nabla_{y} u^{'} ∣^{2} \leq 2 \int (∣ \nabla_{i} u ∣^{2} + ∣ \nabla_{j} u ∣^{2})$ . Both inequalities extend to all $u \in W^{1, 2} (R^{3 N})$ : choose $u_{k} \in C_{c}^{\infty}$ with $u_{k} \to u$ and $\nabla u_{k} \to \nabla u$ in $L^{2}$ ([F3]); the right-hand sides converge to the corresponding norms of $u$ , while the left-hand sides are lower semicontinuous by Fatou along an a.e. convergent subsequence ([F3]).

2.1F1step 1.1algebra

[Operator-norm $K$ -bounds] Let $u \in D (K)$ and put $a : = ∥ K u ∥_{2}$ , $b : = ∥ u ∥_{2}$ . The squared Hardy bound of step 1.1 and [F1] give $∥ u / ∣ x_{i} - a_{m} ∣ ∥_{2}^{2} \leq 4 ∥ \nabla_{i} u ∥_{2}^{2} \leq (32 π^{2} m_{i} / ℏ^{2}) ⟨ K u, u ⟩ \leq (32 π^{2} m_{i} / ℏ^{2}) a b$ . For a pair, the corrected pair bound gives $∥ u / ∣ x_{i} - x_{j} ∣ ∥_{2}^{2} \leq 2 (∥ \nabla_{i} u ∥_{2}^{2} + ∥ \nabla_{j} u ∥_{2}^{2}) \leq (16 π^{2} \max (m_{i}, m_{j}) / ℏ^{2}) ⟨ K u, u ⟩ \leq (16 π^{2} \max (m_{i}, m_{j}) / ℏ^{2}) a b$ : the sum uses $⟨ (K_{i} + K_{j}) u, u ⟩ \leq ⟨ K u, u ⟩$ . For either multiplier let $C$ be the square root of its displayed finite coefficient. For every $η > 0$ , $C \sqrt{a b} \leq η a + C^{2} b / (4 η)$ , obtained by expanding $(\sqrt{η a} - C \sqrt{b} / (2 \sqrt{η}))^{2} \geq 0$ . Thus both actual $L^{2}$ operator norms are at most $η ∥ K u ∥_{2} + C_{η} ∥ u ∥_{2}$ , with finite $C_{η} = C^{2} / (4 η)$ , including $u = 0$ . No form-to-operator-norm inference is used. This proves (a) for the literal Fourier-normalized $K$ of the supplier.

3.1F1step 1.1step 2.1algebra

[Separate quadratic-form bound] The same norm Hardy and kinetic-gradient inequalities give $∥ u / r ∥_{2} \leq C ⟨ K u, u ⟩^{1 / 2}$ for each distance multiplier, so Cauchy–Schwarz gives $∣ ⟨ M_{V} u, u ⟩ ∣ \leq \int ∣ V ∣ ∣ u ∣^{2} \leq C_{V} ⟨ K u, u ⟩^{1 / 2} ∥ u ∥_{2}$ , where $C_{V}$ is the finite sum of the respective Hardy-gradient constants times $∣ z_{i m} ∣, ∣ z_{i j} ∣$ . For every $ϵ > 0$ , Young's square inequality yields $∣ ⟨ M_{V} u, u ⟩ ∣ \leq ϵ ⟨ K u, u ⟩ + C_{V}^{2} ∥ u ∥_{2}^{2} / (4 ϵ)$ . In particular $⟨ (K + M_{V}) u, u ⟩ \geq (1 - ϵ) ⟨ K u, u ⟩ - C_{V}^{2} ∥ u ∥_{2}^{2} / (4 ϵ)$ ; this is a valid form bound and lower estimate, separate from the operator-norm bound of step 2.1.

3.2F5step 2.1algebra

[The full multiplier and Kato-Rellich] Since $V$ is a finite real linear combination of the multipliers of part (a), each with $K$ -bound zero, $V$ is measurable and $M_{V}$ is a symmetric operator with $D (K) \subseteq D (M_{V})$ and $K$ -bound zero ([F5]); in particular the bound $∥ M_{V} u ∥ \leq a ∥ K u ∥ + b ∥ u ∥$ holds on $D (K)$ for some $a < 1$ and finite $b$ (choose $a < 1$ by the zero bound and adjust $b$ ). By [F5] (Kato-Rellich) $H = K + M_{V}$ is self-adjoint on $D (K)$ and bounded below. Since the graph norms of $H$ and $K$ are equivalent on $D (K)$ (as in Bounded fixed-N operator (b)), the core $C_{c}^{\infty}$ of $K$ is a core for $H$ . This proves (b).

4.1step 3.2algebra

[Component extension and permutation sectors] The passage to $L^{2} (R^{3 N}; C) \otimes C^{r}$ is the finite direct sum of $r$ copies of $H$ and is self-adjoint and bounded below by the same argument as part (c) of Bounded fixed-N operator; this proves (c). Under equal masses and permutation-invariant coefficients, both $K$ and the multiplier $M_{V}$ are invariant under the transported permutation unitaries (the weight $κ$ and the multiplier $V$ are permutation invariant, and the domain $D (M_{V})$ is permutation invariant because the a.e.-defined function $V$ is), so $U_{π}^{p h y s} D (H) = D (H)$ and $U_{π}^{p h y s} H = H U_{π}^{p h y s}$ on $D (H)$ ; part (d) then follows from Permutation-restricted self-adjointness (d).

5.1step 1.1step 2.1step 3.2step 4.1algebra∎

[Conclusion and scope] Steps 1.1, 2.1, 3.1, 3.2 and 4.1 prove (a)-(d). The final paragraph of the statement records the excluded claims: no HVZ threshold, no uniform-in- $N$ stability, no thermodynamic limit, no further spectral statement and no physical interpretation is asserted.

Fixed-N Coulomb self-adjointness

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources