Completeness of the Dollard-modified wave operators for the declared Coulomb model

[Numerical SI comparison with dimensionless arguments] Write the fixed SI units as $L_0=1\mathrm{m}$, $T_0=1\mathrm{s}$, $M_0=1\mathrm{k}\mathrm{g}$ and $E_0=1\mathrm{J}$. Let $m_0=m\mathrm{/}{M}_0$, $h_0=\mathrm{\hslash }\mathrm{/}(E_0{T}_0)$, $k_0=\kappa \mathrm{/}(E_0{L}_0)$, $L={\mathrm{\ell }}_{\star }\mathrm{/}{L}_0>0$ and $t_0=t\mathrm{/}{T}_0$ be their numerical values, and put ${\gamma }_0=\gamma L_0=2m_0{k}_0\mathrm{/}{h}_0^2$, $c_0=h_0^2\mathrm{/}(2m_0)$ and $\tau =h_0{t}_0\mathrm{/}(2m_0)$. These are pure real numbers, with $\tau \to \pm \mathrm{\infty }$ as $t\to \pm \mathrm{\infty }$. Represent the wavefunctions in the numerical SI coordinate $q=x\mathrm{/}{L}_0$, with the normalized $L^2$ coordinate map ${\mathcal{J}}_{0}f(q)=L_0^{3\mathrm{/}2}f(L_{0}q)$, the Jacobian normalization understood in the chosen length unit. Substitution in the integral proves this map unitary; weak derivative substitution gives its exact $H^2$ domain transport. The numerical energy representatives are therefore ${\mathsf{H}}_{\kappa }=E_0^{-1}{\mathcal{J}}_0{H}_{\kappa }{\mathcal{J}}_0^{-1}=c_0{C}_{{\gamma }_0}$ and ${\mathsf{H}}_f=E_0^{-1}{\mathcal{J}}_0{H}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}{\mathcal{J}}_0^{-1}=c_0{A}_0$, where $A_0=-{\mathrm{\Delta }}_q$ and $C_{{\gamma }_0}=-{\mathrm{\Delta }}_q-{\gamma }_0\mathrm{/}\mathrm{\mid }q\mathrm{\mid }$. Thus ${\mathcal{J}}_0{e}^{-it{H}_{\kappa }\mathrm{/}\mathrm{\hslash }}{\mathcal{J}}_0^{-1}=e^{-i\tau C_{{\gamma }_0}}$ and similarly for the free operator. The numerical reference time is $t_{\star ,0}=2m_0{L}^2\mathrm{/}{h}_0$, so $\frac{\mathrm{\mid }t\mathrm{\mid }}{t_{\star }}=\frac{\mathrm{\mid }{t}_0\mathrm{\mid }}{t_{\star ,0}}=\frac{\mathrm{\mid }\tau \mathrm{\mid }}{L^2},\log \frac{\mathrm{\mid }t\mathrm{\mid }}{t_{\star }}=\log \mathrm{\mid }\tau \mathrm{\mid }-2\log L\mathrm{.}$ Every logarithm has a dimensionless argument; no time having units of length squared is placed inside the source logarithm.

[Exact fixed energy phases] For a positive numerical energy $e=c_0\lambda$, the phase coefficient of the physical definition is $m_0{k}_0\mathrm{/}(h_0\sqrt{2m_{0}e})={\gamma }_0\mathrm{/}(2\sqrt{\lambda })$. By [F1], $a(P)={\gamma }_0\mathrm{/}(2\mathrm{\mid }P\mathrm{\mid })$ for the numerical source $A_0=\mathrm{\mid }P{\mathrm{\mid }}^2$. For $\sigma \in \{+1,-1\}$ define the fixed bounded unitary $B_{\sigma ,L}=b_{\sigma ,L}(A_0),b_{\sigma ,L}(\lambda )=\exp (-2i\sigma \frac{{\gamma }_0}{2\sqrt{\lambda }}\log L)(\lambda >0),b_{\sigma ,L}(0)=1.$ The functions are bounded unimodular Borel functions despite their oscillation near zero, so their calculus is defined on all $L^2$ and they commute with $e^{-ir{A}_0}$ for every $r$. Spectral correspondence [F3] and step 1.1 give, for $\mathrm{\mid }\tau \mathrm{\mid }\ge 1$ and $\sigma =\mathrm{sign}t$, ${\mathcal{J}}_0{U}_D(t){\mathcal{J}}_0^{-1}=e^{-i\tau A_0}{e}^{i\sigma a(P)\log \mathrm{\mid }\tau \mathrm{\mid }}{B}_{\sigma ,L}=U_D^{\mathrm{n}\mathrm{d}}(\tau )B_{\sigma ,L}\mathrm{.}$ The bounded functional calculus also transports under ${\mathcal{J}}_0$: conjugating the PVM gives a regular PVM representing the numerical operator, so spectral uniqueness identifies it and simple-function approximation gives the claimed calculus transport. If desired, the annular smooth momentum class in [F1] is preserved by each $B_{\sigma ,L}$ because its multiplier is smooth on every compact annulus; hence the source remainder estimate remains integrable on that class after this fixed phase. The comparison for finite $\mathrm{\mid }\tau \mathrm{\mid }<1$ need not equal the source's short-time extension and is irrelevant to the two asymptotic limits.

[Strong limits, isometry and intertwining] Apply [F2] at the positive numerical coupling ${\gamma }_0$. Let $W_{\pm }$ be its two full-space Dollard wave operators. Step 2.1 makes the physical comparison product, conjugated by ${\mathcal{J}}_0$, equal $e^{i\tau C_{{\gamma }_0}}{U}_D^{\mathrm{n}\mathrm{d}}(\tau )B_{\pm ,L}$ on the corresponding exterior half-line. The fixed bounded right factor and the positive scalar time change let the strong limits pass on every vector, giving ${\mathrm{\Omega }}_{D\pm }={\mathcal{J}}_0^{-1}{W}_{\pm }{B}_{\pm ,L}{\mathcal{J}}_0\mathrm{.}$ Both $W_{\pm }$ are isometries by [F2] and both $B_{\pm ,L}$ are unitaries, so these operators are isometries. For physical $s\in \mathbb{R}$, put $r=h_0(s\mathrm{/}{T}_0)\mathrm{/}(2m_0)$. The numerical free/Coulomb flows are $e^{-ir{A}_0}$ and $e^{-ir{C}_{{\gamma }_0}}$ by step 1.1. Since $B_{\pm ,L}$ commute with the free flow and $W_{\pm }{e}^{-ir{A}_0}=e^{-ir{C}_{{\gamma }_0}}{W}_{\pm }$, conjugating gives the exact intertwining of (a). No new source existence or momentum-domain assumption was introduced.

[Full completeness under the reference convention] Because $B_{\pm ,L}$ are onto, step 3.1 gives $\mathrm{ran}{\mathrm{\Omega }}_{D\pm }={\mathcal{J}}_0^{-1}\mathrm{ran}{W}_{\pm }$. By [F2], this is ${\mathcal{J}}_0^{-1}{H}_{\mathrm{a}\mathrm{c}}(C_{{\gamma }_0})={\mathcal{J}}_0^{-1}{H}_{\mathrm{c}}(C_{{\gamma }_0})$. Positive energy scaling by $c_0$ preserves these spectral subspaces [F3], and the unitary numerical coordinate representation transports the spectral measures and their subspaces as in step 2.1. The resulting physical subspaces are exactly $H_{\mathrm{a}\mathrm{c}}(H_{\kappa })=H_{\mathrm{c}}(H_{\kappa })$. This proves (b) for every declared reference length, preserving the source's whole continuous subspace and all angular channels.

[Scattering phases, unitarity and free-flow commutation] Put $S^{\mathrm{n}\mathrm{d}}=W_{+}^{\ast }{W}_{-}$. The formula in step 3.1 gives $S_D={\mathcal{J}}_0^{-1}{B}_{+,L}^{\ast }{S}^{\mathrm{n}\mathrm{d}}{B}_{-,L}{\mathcal{J}}_0\mathrm{.}$ All factors are unitary by [F2] and step 2.1, so $S_D$ is unitary. The commutation with the free flow follows directly from the two intertwinings: taking the adjoint of the plus identity at time $-r$ gives $e^{-ir{A}_0}{W}_{+}^{\ast }=W_{+}^{\ast }{e}^{-ir{C}_{{\gamma }_0}}$, hence $S^{\mathrm{n}\mathrm{d}}{e}^{-ir{A}_0}=W_{+}^{\ast }{e}^{-ir{C}_{{\gamma }_0}}{W}_{-}=e^{-ir{A}_0}{S}^{\mathrm{n}\mathrm{d}}$. The fixed $B_{\pm ,L}$ commute with that same flow, so their scattering product does too, and conjugating proves (c). Equivalently the common range projection from step 4.1 gives both $S_D^{\ast }{S}_D=I$ and $S_D{S}_D^{\ast }=I$. For any other reference time $t_{\star }^{\mathrm{\prime }}$, let $Q_{\sigma }=q_{\sigma }(H^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}})$ with $q_{\sigma }(E)=\exp (-i\sigma \alpha (E)\log (t_{\star }^{\mathrm{\prime }}\mathrm{/}{t}_{\star }))$, $\alpha (E)=m\kappa \mathrm{/}(\mathrm{\hslash }\sqrt{2mE})$ for $E>0$ and $q_{\sigma }(0)=1$. The defining phases give exactly ${\mathrm{\Omega }}_{D\pm }^{\mathrm{\prime }}={\mathrm{\Omega }}_{D\pm }{Q}_{\pm }$ and $S_D^{\mathrm{\prime }}=Q_{+}^{\ast }{S}_D{Q}_{-}$. These incoming/outgoing factors are different sign conventions, not a claim that the scattering operator itself is unchanged.

[Scope and assembly] Steps 3.1–5.1 establish all the original modified-limit, isometry, intertwining, full completeness and scattering-unitarity conclusions with their corrected dimensional convention. No step supplies the ordinary unmodified Coulomb wave limits, so (d) retains exactly that distinction. The reference length adds only a comparison convention and its fixed channel phases; it adds no hypothesis on a state or on the Coulomb interaction, no amplitude formula and no measured cross section.