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Coulomb infinitesimal kinetic bound

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 $γ > 0$ , let $H = L^{2} (R^{3}; C)$ (Hilbert space, The space $L^{p} (μ)$ as the quotient by null functions), and let $H_{0} : = - Δ$ be the free operator of Real multiplication operators and free plus bounded-potential Schrödinger operators with domain $D (H_{0}) = W^{2, 2} (R^{3}; C)$ (Integer-order Sobolev spaces and their norms): $H_{0} = F_{2}^{*} M_{4 π^{2} ∣ ξ ∣^{2}} F_{2}$ and $H_{0} u = - Δ u$ for $u \in D (H_{0})$ , where $Δ u$ is the distributional Laplacian. Write $M$ for the multiplication operator $M u : = \frac{u}{∣ \cdot ∣}, D (M) : = {u \in H : \frac{u}{∣ \cdot ∣} \in H},$ and write $∥ u ∥_{2} = ∥ u ∥_{H}$ . Then:

(a) for every $u \in D (H_{0}) = W^{2, 2} (R^{3}; C)$ one has $u / ∣ \cdot ∣ \in L^{2}$ and

$∥ \frac{u}{∣ \cdot ∣} ∥_{2} \leq 2 \sqrt{∥ u ∥_{2} ∥ Δ u ∥_{2}}, ∥ \frac{u}{∣ \cdot ∣} ∥_{2} \leq η ∥ Δ u ∥_{2} + \frac{1}{η} ∥ u ∥_{2} (η > 0);$

(b) $M$ is a self-adjoint operator on $H$ with $D (H_{0}) \subseteq D (M)$ , symmetric on $D (H_{0})$ , with $H_{0}$ -bound zero in the sense that for every $η > 0$ the pair $(η, η^{- 1})$ is admissible for $M$ relative to $H_{0}$ ;

(c) for every $γ > 0$ the operator $C : = H_{0} - γ M, D (C) : = D (H_{0}) = W^{2, 2} (R^{3}; C),$ is self-adjoint on $H$ , bounded below, and acts by $C u = - Δ u - γ u / ∣ \cdot ∣$ for every $u \in D (C)$ ; for every $η \in (0, 1 / γ)$ one has $C \geq - γ η^{- 1} / (1 - γ η)$ in the quadratic-form sense.

No sharp lower bound of $C$ is claimed here, and no physical interpretation of $C$ , $H_{0}$ , $γ$ or $Δ$ is asserted.

Facts & Assumptions

Given: the free operator $H_{0} = - Δ$ with $D (H_{0}) = W^{2, 2} (R^{3}; C)$ ; the multiplier $M$ ; the constant $γ > 0$ ; a function $u$ of the stated regularity.

[F1]

$H_{0} = F_{2}^{*} M_{4 π^{2} ∣ ξ ∣^{2}} F_{2}$ is self-adjoint on $D (H_{0}) = {f : ∣ ξ ∣^{2} F_{2} f \in L^{2}} = W^{2, 2} (R^{3}; C)$ as a set with equivalent norms, $H_{0} \geq 0$ in the quadratic-form sense, and $H_{0} f = - Δ f$ for Schwartz $f$ (Real multiplication operators and free plus bounded-potential Schrödinger operators, Integer-order W^{k,2} and H^k agree with equivalent norms, Integer-order Sobolev spaces and their norms).

[F2]

For every $u \in W^{1, 2} (R^{3}; C)$ one has $u / ∣ \cdot ∣ \in L^{2}$ and $∥ u / ∣ \cdot ∣ ∥_{2} \leq 2 ∥ \nabla u ∥_{2}$ (Three-dimensional Hardy bound).

[F3]

Plancherel: $F_{2}$ is unitary and for the weak derivatives of $u \in W^{2, 2}$ the multipliers satisfy $F_{2} (\partial_{i} u) (ξ) = 2 π i ξ_{i} F_{2} u (ξ)$ and $F_{2} (Δ u) (ξ) = - 4 π^{2} ∣ ξ ∣^{2} F_{2} u (ξ)$ a.e.; equivalently $∥ \nabla u ∥_{2}^{2} = \int 4 π^{2} ∣ ξ ∣^{2} ∣ F_{2} u ∣^{2}$ and $∥ Δ u ∥_{2}^{2} = \int 16 π^{4} ∣ ξ ∣^{4} ∣ F_{2} u ∣^{2}$ (Plancherel theorem, Distributional derivatives are polynomial Fourier multipliers, Integer-order W^{k,2} and H^k agree with equivalent norms).

[F4]

Cauchy-Schwarz: $\int ∣ f g ∣ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ (Cauchy-Schwarz inequality for $L^{2}$ ).

[F5]

Kato-Rellich: if $A$ is self-adjoint, $B$ symmetric, $D (A) \subseteq D (B)$ and the $A$ -bound of $B$ is less than one with admissible pair $(a, b)$ , then $A + B$ on $D (A)$ is self-adjoint and, when $A \geq γ_{0}$ , satisfies $A + B \geq γ_{0} - \max {a ∣ γ_{0} ∣ + b, b / (1 - a)}$ (Kato-Rellich theorem, Relative boundedness with respect to an operator).

[F6]

If $a$ is a real finite-valued measurable function on $R^{3}$ , then $M_{a} f : = a f$ on ${f \in H : a f \in H}$ is self-adjoint, and it is symmetric on every subspace on which both sides of the pairing are integrable, in particular on the set of $f$ with $a f \in H$ (Real multiplication operators and free plus bounded-potential Schrödinger operators (a), Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators).

Proof

1.1F3F4algebra

[Kinetic inequality] Let $u \in W^{2, 2} (R^{3}; C)$ and put $g : = F_{2} u$ . By [F3], $∥ \nabla u ∥_{2}^{2} = \int 4 π^{2} ∣ ξ ∣^{2} ∣ g (ξ) ∣^{2} d ξ$ and $∥ Δ u ∥_{2}^{2} = \int (4 π^{2} ∣ ξ ∣^{2})^{2} ∣ g (ξ) ∣^{2} d ξ$ , while $∥ u ∥_{2}^{2} = \int ∣ g ∣^{2}$ . Writing $4 π^{2} ∣ ξ ∣^{2} ∣ g ∣^{2} = (∣ g ∣) (4 π^{2} ∣ ξ ∣^{2} ∣ g ∣)$ and applying [F4] to the two factors gives $∥ \nabla u ∥_{2}^{2} \leq ∥ u ∥_{2} ∥ Δ u ∥_{2}$ .

2.1step 1.1F2algebra

[Infinitesimal Coulomb bound] Combining step 1.1 with [F2] yields $∥ u / ∣ \cdot ∣ ∥_{2} \leq 2 ∥ \nabla u ∥_{2} \leq 2 \sqrt{∥ u ∥_{2} ∥ Δ u ∥_{2}}$ . For $η > 0$ the arithmetic-geometric mean inequality $2 \sqrt{a b} \leq η a + η^{- 1} b$ applied to $a = ∥ Δ u ∥_{2}$ , $b = ∥ u ∥_{2}$ gives $2 \sqrt{∥ u ∥_{2} ∥ Δ u ∥_{2}} \leq η ∥ Δ u ∥_{2} + η^{- 1} ∥ u ∥_{2}$ . This proves (a).

3.1step 2.1F1F3F6algebra

[Relative bound of the multiplier] By [F6] applied to $a (x) = 1 / ∣ x ∣$ (finite-valued on $R^{3} ∖ {0}$ and arbitrary, say $0$ , at the origin, a null set) the operator $M$ is self-adjoint on $D (M) = {f : f / ∣ \cdot ∣ \in H}$ , and $M$ is symmetric on $D (M)$ . Step 2.1 shows $D (H_{0}) = W^{2, 2} \subseteq D (M)$ and $∥ M u ∥_{2} = ∥ u / ∣ \cdot ∣ ∥_{2} \leq η ∥ Δ u ∥_{2} + η^{- 1} ∥ u ∥_{2} = η ∥ H_{0} u ∥_{2} + η^{- 1} ∥ u ∥_{2}$ for every $u \in D (H_{0})$ , since $Δ u = - H_{0} u$ on $D (H_{0})$ by [F1] and [F3]. Hence $(η, η^{- 1})$ is an admissible pair for $M$ relative to $H_{0}$ , and the $H_{0}$ -bound of $M$ is zero; this is (b).

4.1step 3.1F1F3F5algebra∎

[Self-adjointness of the Coulomb operator] Fix $γ > 0$ and let $η \in (0, 1 / γ)$ . Then $B : = - γ M$ is symmetric with $D (H_{0}) \subseteq D (B)$ and admissible pair $(a, b) = (γ η, γ η^{- 1})$ with $a < 1$ ; indeed $∥ B u ∥_{2} = γ ∥ M u ∥_{2} \leq γ η ∥ H_{0} u ∥_{2} + γ η^{- 1} ∥ u ∥_{2}$ by step 3.1. By [F5] with $A = H_{0} \geq 0$ (so $γ_{0} = 0$ ) the operator $C = H_{0} + B$ on $D (H_{0})$ is self-adjoint with $C \geq - \max {0, b / (1 - a)} = - γ η^{- 1} / (1 - γ η)$ . Moreover $C u = H_{0} u - γ u / ∣ \cdot ∣ = - Δ u - γ u / ∣ \cdot ∣$ for $u \in D (H_{0})$ , by [F1] and [F3]; this is (c).

Coulomb infinitesimal kinetic bound

Statement

Facts & Assumptions

Proof

Depends on

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