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Controlled quartic-model ground-energy bound

Physical scope

One real mode with a positive quartic polynomial at a fixed finite regulator. The energy bound is a model prediction in joules, not an observed value or a continuum result.

Statement

Assume Countable Choice and use one fixed one-mode regulator of Adopted finite-regulator polynomial model with dimensionless $a > 0$ , $r > 0$ , and $λ \geq 0$ . Here $r$ is a polynomial coefficient, not an SI rest mass. Take $V_{λ} (q) = \frac{r^{2}}{2} q^{2} + λ q^{4}$ on $H_{1} = L^{2} (R, d q; C)$ , with the form realization from Self-adjoint unbounded polynomial interaction at fixed regulator. Let $ϵ_{g} (λ)$ be its least eigenvalue. The physical ground-energy prediction for the adopted scale $E_{*} > 0$ joules is $E_{g}^{p h y s} (λ) : = E_{*} ϵ_{g} (λ)$ . Then

$0 \leq E_{g}^{p h y s} (λ) - E_{g}^{p h y s} (0) \leq \frac{3 E_{*} a λ}{2 r^{2}} .$

For $λ > 0$ , the potential has a nonzero degree-four term and is interacting in the finite-coordinate sense defined by Adopted finite-regulator polynomial model. This conclusion makes no claim about field scattering or a continuum field theory. The inequality is an exact fixed-regulator model deduction; it supplies neither an observed energy nor a perturbation-series or continuum-limit claim.

Facts & Assumptions

Given: Countable Choice; $a > 0$ , $r > 0$ , $λ \geq 0$ ; the one-mode form realizations; and $E_{*} > 0$ joules.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every countable family of nonempty sets has a choice function. The QFT-IR01 spectral and variational suppliers assume this principle.

[A2]

Adopted finite-regulator polynomial model: $q$ is dimensionless, $E_{*} > 0$ is a primitive energy scale in joules, and a self-adjoint form realization $h$ gives $H_{p h y s} = E_{*} h$ . It defines a polynomial interaction term as a nonzero monomial of total degree at least three in $V$ .

[A3]

Physical interpretation of a supplied constructed quantum model: physical interpretation of a Hamiltonian as the selected model's energy operator is adopted only after that self-adjoint operator has actually been constructed.

[F1]

Coercive polynomial form completion and injection: for a real polynomial $V$ with $V (q) \geq c q^{2} - C_{0}$ , the form domain is the completion of $C_{c}^{\infty} (R; C)$ in the norm $∥ u ∥_{Q}^{2} = a ∥ u^{'} ∥_{2}^{2} + \int (V + C_{0} + 2) ∣ u ∣^{2} d q$ , and it injects into $L^{2}$ .

[F2]

Anisotropic Hermite oscillator and quadratic form basis: for $N = 1$ , coefficients $a > 0, c > 0$ , the oscillator form with potential $c q^{2}$ has normalized ground function $g_{b} (q) = (b / π)^{1 / 4} e^{- b q^{2} / 2}$ , $b = \sqrt{c / a}$ , ground eigenvalue $\sqrt{a c}$ , and the proof supplies smooth compact cutoffs with derivative bounded by $C / R$ .

[F3]

Ground eigenvalue as the form variational infimum: under Countable Choice, the least eigenvalue of each selected coercive form realization is the infimum of its form values on normalized vectors in its form domain.

[F4]

Self-adjoint unbounded polynomial interaction at fixed regulator: under Countable Choice and the coercive polynomial hypotheses, the selected form realization is self-adjoint with compact resolvent, and its lowest eigenvalue is attained.

Proof

technique · direct

1.1A1A2F1F4givenalgebra

Since $V_{λ} (q) = r^{2} q^{2} / 2 + λ q^{4} \geq (r^{2} / 2) q^{2}$ , the coercivity hypotheses of [F1] and [F4] hold with $c = r^{2} / 2$ and $C_{0} = 0$ . Countable Choice [A1] therefore gives the form domains of [F1] and the self-adjoint compact-resolvent form realizations with attained least eigenvalues from [F4], for both $q_{λ}$ and $q_{0}$ . If $λ > 0$ , the term $λ q^{4}$ is an interaction term by the finite-coordinate definition in [A2].

2.1A1F1F3F4step 1.1algebra

On $C_{c}^{\infty}$ , the shifted form norms obey $∥ u ∥_{Q_{0}}^{2} \leq ∥ u ∥_{Q_{λ}}^{2}$ because their difference is $λ \int q^{4} ∣ u ∣^{2} d q \geq 0$ . Hence the identity extends continuously $Q_{λ} \to Q_{0}$ , and the two continuous injections into $L^{2}$ identify the same representative. On the core, $q_{λ} (u) = q_{0} (u) + λ \int q^{4} ∣ u ∣^{2} d q \geq q_{0} (u)$ ; approximating any $u \in Q_{λ}$ by core elements and using continuity of both forms passes this inequality to $u$ . For a normalized $u$ , [F3] gives $q_{0} (u) \geq ϵ_{g} (0)$ , so taking the infimum on $Q_{λ}$ yields $ϵ_{g} (λ) \geq ϵ_{g} (0)$ .

2.2A1F1F2F4step 1.1constructalgebra

Let $g_{b} (q) = (b / π)^{1 / 4} e^{- b q^{2} / 2}$ with $b = r / \sqrt{2 a}$ , the normalized ground function in [F2] for $c = r^{2} / 2$ . Choose the smooth cutoff $χ_{R} (q) = χ (q / R)$ from [F2], with $χ = 1$ on $[- 1, 1]$ , $χ = 0$ outside $[- 2, 2]$ , and $∣ χ_{R}^{'} ∣ \leq C / R$ . Then $u_{R} = χ_{R} g_{b} \in C_{c}^{\infty}$ . For $S > R$ , $u_{R} - u_{S}$ is supported where $∣ q ∣ \geq R$ , and $∥ (u_{R} - u_{S})^{'} ∥_{2} \leq ∥ 1_{∣ q ∣ \geq R} g_{b}^{'} ∥_{2} + C (R^{- 1} + S^{- 1}) ∥ 1_{∣ q ∣ \geq R} g_{b} ∥_{2} \to 0$ . The weighted part of its $Q_{λ}$ norm is bounded in square by $\int_{∣ q ∣ \geq R} (r^{2} q^{2} / 2 + λ q^{4} + 2) ∣ g_{b} (q) ∣^{2} d q \to 0$ . These tails vanish because each polynomial times $e^{- b q^{2}}$ is integrable: for large $∣ q ∣$ , its product with $∣ q ∣^{2}$ is bounded. Thus $(u_{R})$ is $Q_{λ}$ -Cauchy. Its $L^{2}$ limit is $g_{b}$ , so injectivity in [F1] gives $g_{b} \in Q_{λ}$ .

3.1A1F2step 2.2algebra

From $g_{b}^{'} = - b q g_{b}$ and $∥ g_{b} ∥_{2} = 1$ , integrate the derivatives of $q e^{- b q^{2}}$ and $q^{3} e^{- b q^{2}}$ over $R$ . Their boundary terms vanish by Gaussian decay, giving $\int q^{2} ∣ g_{b} ∣^{2} d q = 1 / (2 b)$ and $\int q^{4} ∣ g_{b} ∣^{2} d q = 3 / (4 b^{2})$ . Hence $q_{λ} (g_{b}) = a b / 2 + r^{2} / (4 b) + 3 λ / (4 b^{2})$ . At $b = r / \sqrt{2 a}$ , the first two terms equal $ϵ_{g} (0) = r \sqrt{a / 2}$ by [F2], and the last term is $3 a λ / (2 r^{2})$ .

4.1A1F3step 2.1step 3.1algebra

Since $g_{b} \in Q_{λ}$ is normalized, [F3] gives $ϵ_{g} (λ) \leq q_{λ} (g_{b})$ . Combining this with step 2.1 and step 3.1 yields $0 \leq ϵ_{g} (λ) - ϵ_{g} (0) \leq 3 a λ / (2 r^{2})$ .

5.1A2A3step 4.1algebra∎

Multiplication by $E_{*} > 0$ preserves both inequalities and gives the stated joule-valued bound. By [A3], this is a selected-model energy prediction and not a reported measurement.

Controlled quartic-model ground-energy bound

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources