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The slowly varying one-centre Coulomb source: verified uniform gap of the n-th level and the 1/T leakage estimate

Physical scope

The declared one-centre Coulomb model with a prescribed C^2 path of the external source strength (the coupling kappa): the identification of the n-th Coulomb level and of its n^2-fold eigenspace along the path, the verification of a uniform positive gap from the model's spectrum theorem with the explicit rational lower bound, and the operator-norm and Born leakage estimates for an initial preparation in that level under the slowly varying source. The theorem concerns the declared external-source model only; it asserts no laboratory field or control, no linear Stark coupling, no transition rate, no branch selection and no empirical result.

Statement

Adopt the declared one-centre Coulomb model of The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions clauses (a)-(c) with mass parameter $m > 0$ , action constant $ℏ > 0$ and declared coupling $κ > 0$ of unit $J m$ , and its energy operator $H_{κ} = \frac{ℏ^{2}}{2 m} C_{γ}$ , $C_{γ} = - Δ - γ / ∣ \cdot ∣$ on $Y = H^{2} (R^{3}; C)$ with $γ = 2 m κ / ℏ^{2}$ . Adopt the declared adiabatic protocol of The declared adiabatic protocol model: slow parameter, followed spectral subspace, uniform gap and leakage probability in its common-domain branch (u) with the following declared source-strength family: a prescribed path $κ \in C^{2} ([0, 1]; (0, \infty))$ with $0 < κ_{\min} \leq κ (s) \leq κ_{\max} < \infty$ , the family $H (s) : = H_{κ (s)}, γ (s) : = \frac{2 m κ (s)}{ℏ^{2}}, A (s) : = - Δ - \frac{γ (s)}{∣ \cdot ∣}, D (A (s)) = Y,$ read as the model's energy at slow time $s$ , a protocol duration $T > 0$ (seconds), and the protocol propagator $U_{T}$ defined by $i ℏ \partial_{s} U_{T} (s, σ) = T H (s) U_{T} (s, σ)$ , $U_{T} (σ, σ) = I$ . For this prescribed positive $C^{2}$ path, the exact common domain, uniform graph-norm equivalence and $C^{2} ([0, 1]; B (Y, L^{2}))$ regularity are consequences of the Coulomb bound, proved below; they require no additional admissibility assumption on the path. Fix $n \geq 1$ and let $P_{n} (s)$ be the spectral projection of $H (s)$ for its $n$ -th level. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Then:

(a) Level, degeneracy and uniform gap. For every $s \in [0, 1]$ the number $E_{n} (s) = - \frac{m κ (s)^{2}}{2 ℏ^{2} n^{2}} = \frac{ℏ^{2}}{2 m} (- \frac{γ (s)^{2}}{4 n^{2}})$ is an eigenvalue of $H (s)$ with eigenspace of dimension $n^{2}$ , the spectrum of $H (s)$ has no eigenvalues in $[0, \infty)$ , its essential spectrum is $[0, \infty)$ , and the complementary distance of the level is bounded below uniformly by $dist (E_{n} (s), σ (H (s)) ∖ {E_{n} (s)}) \geq \frac{ℏ^{2}}{2 m} δ_{n} γ_{\min}^{} = : g_{n} > 0, δ_{n} : = \frac{2 n + 1}{4 n^{2} (n + 1)^{2}}, γ_{\min} = \frac{2 m κ_{\min}}{ℏ^{2}} .$ Consequently $P_{n} \in C^{2} ([0, 1]; B (L^{2}, Y))$ , and the family of projections $s \mapsto P_{n} (s)$ is a uniformly gapped, coherently followed label of the protocol in the sense of The declared adiabatic protocol model: slow parameter, followed spectral subspace, uniform gap and leakage probability clause (c).

(b) Leakage estimate. There is a constant $C_{n} < \infty$ , depending only on $n$ , on the mass parameter $m$ , on $ℏ$ and on $κ_{\min}, κ_{\max}$ and the $C^{2}$ norm of the path $κ$ (that is, on the data of the declared family), such that for every $s \in [0, 1]$ $∥ (I - P_{n} (s)) U_{T} (s, 0) P_{n} (0) ∥ \leq \frac{C_{n}}{T},$ and therefore, for every unit vector $ψ (0) \in ran P_{n} (0)$ with $ψ (t) = U_{T} (t / T, 0) ψ (0)$ , $p_{l e a k} (s) = ∥ (I - P_{n} (s)) ψ (s T) ∥^{2} \leq \frac{C_{n}^{2}}{T^{2}}, ⟨ P_{n} (s) ψ (s T), ψ (s T) ⟩ \geq 1 - \frac{C_{n}^{2}}{T^{2}} .$ The same Born leakage and survival bounds hold for every supported mixed preparation $ρ_{0} = P_{n} (0) ρ_{0} P_{n} (0)$ , with $ρ (s) = U_{T} (s, 0) ρ_{0} U_{T} (s, 0)^{*}$ and $p_{l e a k} (s) = tr (ρ (s) (I - P_{n} (s)))$ .

(c) Scope. The theorem concerns the declared external-source family only: the coupling path is a prescribed source strength of the one-centre model, not an applied laboratory field, and the statement asserts no laboratory control, no linear Stark coupling (which has no uniformly gapped branch), no transition rate, no branch selection or relaxation, and no empirical result.

Facts & Assumptions

Given: the declared one-centre Coulomb model with parameters $m, ℏ, κ$ and the source-strength path $κ (\cdot)$ ; the family $H (s) = H_{κ (s)}$ , $γ (s) = 2 m κ (s) / ℏ^{2}$ , $A (s)$ ; the duration $T$ and propagator $U_{T}$ ; the label $n \geq 1$ and projections $P_{n} (s)$ ; the Axiom of Countable Choice and the Axiom of Choice.

[F1]

Coulomb model data. $H_{κ} = \frac{ℏ^{2}}{2 m} C_{γ}$ is self-adjoint on the exact domain $Y = H^{2} (R^{3}; C)$ (Integer-order Sobolev spaces and their norms) for every $κ > 0$ , with $γ = 2 m κ / ℏ^{2}$ , and $(H_{κ}, J)$ is the model's declared energy observable with length scale $a_{κ} = ℏ^{2} / (m κ)$ and energy scale $ℏ^{2} / (2 m a_{κ}^{2})$ (The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions clauses (a)-(c)).

[F2]

Coulomb spectrum. For the fixed coupling $κ$ : $Ψ_{n l m} \in D (H_{κ})$ are unit eigenvectors with $H_{κ} Ψ_{n l m} = E_{n} Ψ_{n l m}$ , $E_{n} = - m κ^{2} / (2 ℏ^{2} n^{2})$ ; the eigenvalues of $H_{κ}$ are exactly the numbers $E_{n}$ , $n \geq 1$ , the eigenspace at $E_{n}$ has dimension $\sum_{l = 0}^{n - 1} (2 l + 1) = n^{2}$ , and $H_{κ}$ has no eigenvalues in $[0, + \infty)$ , whose spectral content is entirely essential spectrum $[0, + \infty)$ (The one-centre Coulomb model: bound energies, n^2-degeneracy, completeness of the bound states in the negative spectral subspace and the continuum clauses (a)-(c), together with the essential-spectrum statement in clause (b)). The same statements hold for the scalar multiple $A (s) = \frac{2 m}{ℏ^{2}} H_{κ (s)}$ , with eigenvalues multiplied by $2 m / ℏ^{2}$ .

[F3]

Atomic adiabatic supplier. For $A (s) = - Δ - γ (s) / ∣ \cdot ∣$ on $Y$ with $γ \in C^{2} ([0, 1]; (0, \infty))$ , $0 < γ_{\min} \leq γ (s) \leq γ_{\max} < \infty$ , if $e_{n} (s) = - γ (s)^{2} / (4 n^{2})$ is the named eigenvalue with uniform complementary gap $g > 0$ and $U_{T}$ solves $i \partial_{s} U_{T} = T A (s) U_{T}$ , then there is $C < \infty$ , depending on $n, γ_{\min}, γ_{\max}$ and the $C^{2}$ norm of $γ$ , with $∥ (I - P_{n} (s)) U_{T} (s) P_{n} (0) ∥ \leq C / T$ for $s \in [0, 1]$ (Unbounded gapped atomic adiabatic estimate); the strong propagator $U_{T}$ of the common-domain family exists and is unique (Common-domain nonautonomous evolution).

[F4]

Protocol readings. The protocol declares $U_{T}$ as its propagator, $P_{n} (s)$ as a followed spectral projection when the label is uniformly gapped and coherently followed, and $p_{l e a k} (s) = ∥ (I - P_{n} (s)) ψ (s T) ∥^{2}$ as the Born leakage probability of a unit initial vector in $ran P_{n} (0)$ (The declared adiabatic protocol model: slow parameter, followed spectral subspace, uniform gap and leakage probability clauses (b), (c); Assigned sharp observables and the outcome law of the framework clauses (a), (b); State and composition postulate P1: Hilbert spaces, preparations and composite systems; Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes; The state space: density operators, pure and mixed states, and purity; Dimensions, SI unit conventions and quantity types of the framework).

[F5]

Spectral and norm readings. $σ (H_{κ})$ and the projections are those of Discrete and essential spectrum of a self-adjoint operator and Projection valued measure; for a bounded $B$ and vector $v$ , $∥ B v ∥ \leq ∥ B ∥ ∥ v ∥$ and complementary projections $Q$ , $I - Q$ satisfy $∥ Q v ∥^{2} + ∥ (I - Q) v ∥^{2} = ∥ v ∥^{2}$ (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum, Hilbert space, Symmetric, self-adjoint and essentially self-adjoint operators).

[F6]

Exact-domain Coulomb estimate and graph control. For $B = - Δ$ on $Y = H^{2}$ , $M u = u / ∣ x ∣$ and every $η > 0$ , $∥ M u ∥_{2} \leq η ∥ B u ∥_{2} + η^{- 1} ∥ u ∥_{2}$ ; $B - γ M$ is self-adjoint on exactly $Y$ for every $γ > 0$ (Coulomb infinitesimal kinetic bound clauses (a),(c)). The fixed Fourier/Sobolev norm satisfies $∥ u ∥_{Y} \leq c_{0} (∥ u ∥_{2} + ∥ B u ∥_{2})$ , and $B : Y \to L^{2}$ is bounded; the uniform graph and $C^{2}$ operator estimates are also supplied explicitly by Unbounded gapped atomic adiabatic estimate proof 1.1. Its proof 3.1 supplies the resolvent/PVM contour argument and inverse derivative formulas on the same exact domain, used explicitly below.

[F7]

Supported densities. Every density has a trace-norm convergent orthonormal expansion with positive eigenvalue weights $ρ_{0} = \sum_{j} w_{j} T_{u_{j}, u_{j}}$ , $\sum_{j} w_{j} = 1$ , and its projection probabilities are the weighted pure projection probabilities (Density operators and scalar spectral probability, Statement(a),(b)).

Proof

technique · direct; identify the scaled physical family with the supplier's normalized Coulomb family, derive common-domain regularity, verify the uniform gap and coherent Riesz following from the model's spectrum theorem, and rescale the supplier's $1/T$ estimate to physical time

Given: the data of the Statement and [F1]-[F7].

1.1givenF1F3F6algebra

[Common domain and automatic regularity] Put $B = - Δ$ , $M u = u / ∣ x ∣$ and $γ_{\max} = 2 m κ_{\max} / ℏ^{2}$ . By [F6], each $A (s) = B - γ (s) M$ is self-adjoint on exactly the dense Hilbert domain $Y = H^{2}$ , continuously embedded in $L^{2}$ . Choose $η = (2 γ_{\max})^{- 1}$ . For every $u \in Y$ , $γ (s) ∥ M u ∥_{2} \leq \frac{1}{2} ∥ B u ∥_{2} + 2 γ_{\max}^{} ∥ u ∥_{2}$ , whence $∥ A (s) u ∥_{2} \leq \frac{3}{2} ∥ B u ∥_{2} + 2 γ_{\max}^{} ∥ u ∥_{2}$ and $∥ B u ∥_{2} \leq 2 ∥ A (s) u ∥_{2} + 4 γ_{\max}^{} ∥ u ∥_{2}$ . Together with the fixed Fourier/Sobolev equivalence of [F6], these give uniform equivalence between $∥ u ∥_{Y}$ and $∥ u ∥_{2} + ∥ A (s) u ∥_{2}$ . Also $M : Y \to L^{2}$ is bounded, by the same estimate, so $A \in C^{2} ([0, 1]; B (Y, L^{2}))$ with $A^{'} = - γ^{'} M$ and $A^{''} = - γ^{''} M$ . All these norms are controlled by $γ_{\max}$ and $∥ γ ∥_{C^{2}}$ . Positive multiplication by $ℏ^{2} / (2 m)$ gives the same exact domain and uniform graph equivalence for $H (s)$ and yields $H^{'} = - κ^{'} M$ , $H^{''} = - κ^{''} M$ . Thus every stated positive $C^{2}$ path automatically satisfies branch (u); [F3] supplies the unique domain-preserving unitary propagator for the self-adjoint $C^{1}$ family $T H (s) / ℏ$ on $Y$ .

2.1F1F2F5step 1.1algebra

[Correct adjacent gaps and the whole level] By [F1] and [F2], $H (s) = (ℏ^{2} / (2 m)) A (s)$ has eigenvalues exactly $E_{n} (s) = - (ℏ^{2} / (2 m)) γ (s)^{2} / (4 n^{2})$ , each eigenspace having dimension $n^{2}$ ; its nonnegative spectrum is the essential interval $[0, \infty)$ and contains no eigenvalue. Since $E_{1} < E_{2} < \dots < 0$ , the complementary distance at $E_{n}$ is the minimum of the positive differences $E_{n + 1} - E_{n}$ , $E_{n} - E_{n - 1}$ when $n \geq 2$ , and $∣ E_{n} ∣$ . Dividing by $(ℏ^{2} / (2 m)) γ (s)^{2}$ gives, respectively, $δ_{n} = (2 n + 1) / (4 n^{2} (n + 1)^{2})$ , $(2 n - 1) / (4 n^{2} (n - 1)^{2})$ and $1 / (4 n^{2})$ . The first is the smallest: $2 n + 1 \leq (n + 1)^{2}$ , and $(2 n - 1) (n + 1)^{2} - (2 n + 1) (n - 1)^{2} = 6 n^{2} - 2 > 0$ for $n \geq 2$ . Hence the distance is exactly $(ℏ^{2} / (2 m)) γ (s)^{2} δ_{n} \geq g_{n} > 0$ . For the normalized family $A (s)$ this gives the uniform bound ${\tilde{g}}_{n} = δ_{n} γ_{\min}^{} > 0$ , with $g_{n} = (ℏ^{2} / (2 m)) {\tilde{g}}_{n}$ . The eigenprojection refers to the entire $n^{2}$ -dimensional space; no vector or sublevel is selected.

3.1step 1.1step 2.1F1F2F4F6constructalgebra

[Projection regularity and coherent local Riesz following] Write $e_{n} (s) = - γ (s)^{2} / (4 n^{2})$ , $ρ = {\tilde{g}}_{n} / 3$ and $z (s, θ) = e_{n} (s) + ρ e^{i θ}$ . Step 2.1 gives spectral distance at least $ρ$ on this circle. The exact-domain resolvent $R (s, θ) = (z (s, θ) - A (s))^{- 1}$ therefore has $L^{2}$ norm at most $ρ^{- 1}$ by the contour/PVM argument in [F6]. The identity $A (s) R = z R - I$ and step 1.1's uniform graph bound give a uniform $L^{2} \to Y$ bound, since $z$ is uniformly bounded by the path data. Put $B_{j} (s) = A^{(j)} (s) - e_{n}^{(j)} (s) I : Y \to L^{2}$ for $j = 1, 2$ . The inverse equations on these exact spaces give joint continuity and $C^{2}$ regularity in $s$ by the local Neumann identity; differentiating gives $R^{'} = R B_{1} R$ and $R^{''} = R B_{2} R + 2 R B_{1} R B_{1} R$ in $B (L^{2}, Y)$ . These derivatives are uniformly bounded by the established bounds and $∥ γ ∥_{C^{2}}$ . Thus $P_{n} (s) = (ρ / (2 π)) \int_{0}^{2 π} e^{i θ} R (s, θ) d θ$ is $C^{2}$ in $B (L^{2}, Y)$ ; the PVM contour formula of [F6] identifies this integral with the full eigenprojection, and positive energy scaling leaves that eigenspace and projection unchanged. For any $s_{0}$ , continuity of $e_{n}$ gives a neighborhood with $∣ e_{n} (s) - e_{n} (s_{0}) ∣ < ρ / 2$ . The fixed circle $∣ z - e_{n} (s_{0}) ∣ = ρ$ then encloses $e_{n} (s)$ and excludes the rest of the spectrum, whose distance from its center is at least ${\tilde{g}}_{n} - ρ / 2 > ρ$ . Its resolvents have the same local $C^{2}$ inverse construction, and its Riesz integral is exactly $P_{n} (s)$ . On overlapping neighborhoods the integrals agree as the same named spectral projection. Therefore $P_{n}$ is norm-continuous and coincides with coherent local Riesz continuation from $P_{n} (0)$ in [F4], proving the final assertion of (a).

4.1step 1.1step 2.1step 3.1F1F3algebra

[Rescaling the supplier] By [F1] the family $A (s)$ satisfies the hypotheses of [F3] with $γ_{\min} = 2 m κ_{\min} / ℏ^{2}$ and $γ_{\max} = 2 m κ_{\max} / ℏ^{2}$ , and by step 2.1 the normalized named eigenvalue has the uniform gap ${\tilde{g}}_{n} > 0$ , with coherent $C^{2}$ projection supplied by step 3.1, so [F3] supplies a constant $C < \infty$ with the normalized bound $∥ (I - P_{n} (s)) U_{S}^{A} (s) P_{n} (0) ∥ \leq C / S$ , where $U_{S}^{A}$ solves $i \partial_{s} U_{S}^{A} = S A (s) U_{S}^{A}$ . The physical family is $H (s) = \frac{ℏ^{2}}{2 m} A (s)$ , so the physical equation $i ℏ \partial_{s} U_{T} = T H (s) U_{T}$ is $i \partial_{s} U_{T} = \frac{T ℏ}{2 m} A (s) U_{T}$ ; hence $U_{T} = U_{S}^{A}$ with $S = \frac{T ℏ}{2 m}$ and the normalized bound becomes $∥ (I - P_{n} (s)) U_{T} (s) P_{n} (0) ∥ \leq 2 m C / (ℏ T)$ . Setting $C_{n} : = \frac{2 m C}{ℏ}$ (seconds; finite, depending only on $n, m, ℏ$ and the path data through $γ$ ) gives the operator bound of clause (b).

5.1step 4.1F4F5algebra

[Born leakage and survival] Let $ψ (0) \in ran P_{n} (0)$ be unit, so $P_{n} (0) ψ (0) = ψ (0)$ ; $U_{T} (s, 0)$ is unitary, so $ψ (s T)$ is unit, and $(I - P_{n} (s)) ψ (s T) = (I - P_{n} (s)) U_{T} (s, 0) P_{n} (0) ψ (0)$ ; the operator bound of step 4.1 and [F5] give $∥ (I - P_{n} (s)) ψ (s T) ∥ \leq C_{n} / T$ , whose square is $p_{l e a k} (s)$ by [F4], and the survival probability is $1 - p_{l e a k} (s) \geq 1 - C_{n}^{2} / T^{2}$ by [F5]. This is clause (b).

6.1step 5.1F7algebra

[Supported mixed preparations] Let $ρ_{0} = P_{n} (0) ρ_{0} P_{n} (0)$ . In the expansion of [F7], every vector of positive eigenvalue $w_{j}$ satisfies $u_{j} = w_{j}^{- 1} ρ_{0} u_{j} \in ran P_{n} (0)$ . Unitary transport gives the same weights and orthonormal vectors $U_{T} (s, 0) u_{j}$ in the evolved density. By [F7], its leakage projection probability is $\sum_{j} w_{j} ∥ (I - P_{n} (s)) U_{T} (s, 0) u_{j} ∥^{2}$ . The pure bound in step 5.1 holds for each such $u_{j}$ , so this sum is at most $C_{n}^{2} / T^{2}$ since $\sum_{j} w_{j} = 1$ . The complementary projection probability is $1 - p_{l e a k} (s) \geq 1 - C_{n}^{2} / T^{2}$ . All operators in the trace expression are bounded projections; no unbounded-density product or eigenvector selection within the degenerate energy level is required.

7.1step 1.1step 2.1step 3.1step 4.1step 5.1given∎

[Scope] The identification used only the declared one-centre model, its spectrum theorem and the normalized adiabatic supplier; the coupling path is an external source strength of that model, no laboratory field, Stark coupling, rate, selection or empirical comparison entered, which is clause (c).

Remarks

Where the physics is added. The supplier proves the estimate for the abstract family $A (s) = - Δ - γ (s) / ∣ \cdot ∣$ ; the theorem adds the identification of that family with the declared Coulomb model at a prescribed source strength, the verification of the uniform gap from the model's own spectrum theorem with the explicit rational constant $δ_{n}$ , and the conversion of the supplier's slow-time parameter to the physical protocol duration.
Automatic regularity for the stated path. The Hardy/infinitesimal kinetic bound proves the exact common $H^{2}$ domain, uniform graph-norm equivalence and $C^{2} (H^{2} \to L^{2})$ regularity for every prescribed positive $C^{2}$ source-strength path. The explicit contour construction proves $C^{2}$ projection regularity and coherent Riesz following for the entire $n^{2}$ -dimensional eigenspace. No additional path-admissibility or eigenvector-selection assumption is needed.

The slowly varying one-centre Coulomb source: verified uniform gap of the n-th level and the 1/T leakage estimate

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources