The declared long-range Coulomb model and its Dollard-modified free comparison

(a) Framework data. Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4, the kinematics and massive-particle model postulate P5 and the identical-particle and spin postulate P6 (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, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations), with the spin-zero state space $H_0=L^2({\mathbb{R}}^3;\mathbb{C})$, mass $m>0$, action constant $\mathrm{\hslash }>0$ and the units of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) The declared Coulomb model. As an additional model assumption of the pair, fix a coupling $\kappa >0$, read in $\mathrm{J}\mathrm{m}$, and adopt the one-centre Coulomb model of The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions in its spin-zero one-particle reading: the declared potential is $V_{\kappa }(x):=-\frac{\kappa }{\mathrm{\mid }x\mathrm{\mid }}(x\mathrm{\ne }0),$ read in joules, with the value at the origin a declared convention that does not affect the theory, and the assigned energy observable is $H_{\kappa }=-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\Delta }}_w-\frac{\kappa }{\mathrm{\mid }\cdot \mathrm{\mid }}$ on the domain $H^2({\mathbb{R}}^3;\mathbb{C})=W^{2,2}({\mathbb{R}}^3;\mathbb{C})$, self-adjoint there by The one-centre Coulomb model: energy observable, flow, stationary states and spectral support clause (a) with flow $e^{-it{H}_{\kappa }\mathrm{/}\mathrm{\hslash }}$ in the reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution. The model declares the free comparison operator to be the free assigned energy observable $H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$ of The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units clause (c), and reads its Coulomb coupling in the inverse-length form $\gamma :=\frac{2m\kappa }{{\mathrm{\hslash }}^2}({\mathrm{m}}^{-1}),\text{so that}{H}_{\kappa }=c(-{\mathrm{\Delta }}_w-\frac{\gamma }{\mathrm{\mid }\cdot \mathrm{\mid }}),c=\frac{{\mathrm{\hslash }}^2}{2m}\mathrm{.}$ The pure Coulomb potential is not in the short-range classes of The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units clause (b); it is the model's long-range case.

(c) The declared Dollard-modified comparison. Choose a reference length ${\mathrm{\ell }}_{\star }>0$, read in metres, as a comparison convention, and put $t_{\star }:=\frac{2m{\mathrm{\ell }}_{\star }^2}{\mathrm{\hslash }}(\mathrm{s})\mathrm{.}$ and, for $t\mathrm{\ne }0$, let $h_t:[0,\mathrm{\infty })\to \mathbb{C}$ be the bounded Borel function $h_t(x):=\exp (i\mathrm{sign}(t)\frac{m\kappa }{\mathrm{\hslash }\sqrt{2mx}}\log \frac{\mathrm{\mid }t\mathrm{\mid }}{t_{\star }})(x>0),h_t(0):=1.$ The model declares the Dollard-modified free comparison to be the family $U_D(t):=e^{-it{H}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}\mathrm{/}\mathrm{\hslash }}{h}_t(H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}})(t\mathrm{\ne }0),U_D(0):=I,$ where $h_t(H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}})$ is the bounded functional calculus (Bounded borel pvm integral clause 1) of the self-adjoint free operator: since $h_t$ is unimodular, $U_D(t)$ is unitary for every $t$, and both factors are functions of $H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$ and commute. The modified wave operators of the model are the strong limits ${\mathrm{\Omega }}_{D\pm }:={\mathrm{s-lim}}_{t\to \pm \mathrm{\infty }}{e}^{it{H}_{\kappa }\mathrm{/}\mathrm{\hslash }}{U}_D(t),$ declared when both exist, and the modified scattering operator is $S_D:={\mathrm{\Omega }}_{D+}^{\ast }{\mathrm{\Omega }}_{D-}$ when both exist and are isometries. The reference length and the resulting time $t_{\star }$ are declared conventions. More exactly, changing the reference time from $t_{\star }$ to $t_{\star }^{\mathrm{\prime }}$ multiplies $U_D(t)$ on the right, for each sign $\sigma =\mathrm{sign}t$, by the fixed free-energy unitary $Q_{\sigma }=q_{\sigma }(H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}),q_{\sigma }(E)=\exp (-i\sigma \frac{m\kappa }{\mathrm{\hslash }\sqrt{2mE}}\log \frac{t_{\star }^{\mathrm{\prime }}}{t_{\star }})(E>0),q_{\sigma }(0)=1.$ It is a bounded unimodular Borel multiplier and commutes with the free flow. Hence, when the wave operators exist, ${\mathrm{\Omega }}_{D\pm }^{\mathrm{\prime }}={\mathrm{\Omega }}_{D\pm }{Q}_{\pm }$ and $S_D^{\mathrm{\prime }}=Q_{+}^{\ast }{S}_D{Q}_{-}$. The completeness and isometry properties are unchanged; these are the fixed outgoing and incoming phase conventions.

(d) Status and scope. Clauses (b)-(c) are declarations of the pair: a prescribed long-range potential with its assigned energy observable, and a modified comparison construction for it. The model asserts no formula for a Coulomb scattering amplitude, no Rutherford cross section and no measured quantity; the modification is a comparison construction for the long-range generator, not a new interaction or a measured force. The ordinary short-range comparison of The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units clause (d) is not claimed to be complete for the Coulomb model; its completeness is established only for the modified comparison, on this page.

• Why a modified comparison is needed. The free comparison $e^{it{H}_{\kappa }\mathrm{/}\mathrm{\hslash }}{e}^{-it{H}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}\mathrm{/}\mathrm{\hslash }}$ does not have a complete strong limit in general for a $1\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$ tail: the classical phase mismatch of the long-range interaction is not integrable in time. The logarithmic phase of $h_t$ is the declared correction; the model does not derive it from a measured force law.
• Units of the phase. The exponent is dimensionless: $m\kappa \mathrm{/}(\mathrm{\hslash }\sqrt{2mx})$ is dimensionless for $x$ in joules, and $\log (\mathrm{\mid }t\mathrm{\mid }\mathrm{/}{t}_{\star })$ is dimensionless for $t$ in seconds and $t_{\star }=2m{\mathrm{\ell }}_{\star }^2\mathrm{/}\mathrm{\hslash }$ in seconds. Indeed $m\mathrm{/}\mathrm{\hslash }$ has dimension $\mathrm{s}{\mathrm{m}}^{-2}$, and the reference length squared supplies the missing ${\mathrm{m}}^2$. The quantity $\gamma =2m\kappa \mathrm{/}{\mathrm{\hslash }}^2$ has inverse-length units; its numerical coefficient becomes dimensionless only after a length unit is fixed.