Dollard channel completeness

The Hilbert spaces below have fixed countable dense sequences of rational complex simple functions on rational boxes or intervals. Their density follows by simple-function approximation and Lebesgue regularity; CC selects countable approximation sequences for each fixed input. Closed subspaces use their orthogonal projections of these sequences. For a real measurable multiplier $M_a$, truncation to $\mathrm{\mid }a\mathrm{\mid }\le n$ proves dense domain, and testing its adjoint against all vectors supported in $\mathrm{\mid }a\mathrm{\mid }\le n$ identifies the adjoint as $M_a$ on exactly $\{h:ah\in L^2\}$. Thus the explicit Fourier multiplier gives self-adjoint $H_0$ on $H^2$. Hardy's inequality follows by integrating $\mathrm{div}(x\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2)\mathrm{\mid }u{\mathrm{\mid }}^2$ for smooth $u$ away from zero and using Cauchy–Schwarz: $\mathrm{\parallel }u\mathrm{/}r{\mathrm{\parallel }}_2^2\le 2\mathrm{\parallel }u\mathrm{/}r{\mathrm{\parallel }}_2\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2$. Smooth cutoffs remove the puncture, and Sobolev density extends $\mathrm{\parallel }u\mathrm{/}r{\mathrm{\parallel }}_2\le 2\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2$ to $H^1$. Fourier Cauchy–Schwarz gives $\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2^2\le \mathrm{\parallel }u{\mathrm{\parallel }}_2\mathrm{\parallel }{H}_{0}u{\mathrm{\parallel }}_2$. For $z=\pm i\mu$, the explicit $R_0(z)=(H_0-z{)}^{-1}$ has $\mathrm{\parallel }{R}_0\mathrm{\parallel }\le {\mu }^{-1}$ and $\mathrm{\parallel }{H}_0{R}_0\mathrm{\parallel }\le 1$, hence $\mathrm{\parallel }\gamma r^{-1}{R}_0(z)\mathrm{\parallel }\le 2\gamma \mathrm{/}\sqrt{\mu }<1$ when $\mu >4{\gamma }^2$. The factorization $C-z=(I-\gamma r^{-1}{R}_0(z))(H_0-z)$, $C=H_0-\gamma \mathrm{/}r$ on $H^2$, gives onto-ness for both signs by the norm-convergent Neumann inverse. For $y\in D(C^{\ast })$ solve $(C-i\mu )x=(C^{\ast }-i\mu )y$; then $y-x\in \ker (C^{\ast }-i\mu )=\mathrm{ran}(C+i\mu {)}^{\perp }=0$. Thus $C=C^{\ast }$ on exactly $H^2$. Also $⟨Cu,u⟩\ge -{\gamma }^2\mathrm{\parallel }u{\mathrm{\parallel }}_2^2$. This uses the explicit Hardy and Fourier calculation, rather than general Kato–Rellich.

Apply CC spectral calculus on a Hilbert space with a fixed countable dense sequence to $H_0,C$ from step 1.1 and to the locally constructed half-line operators below, using their fixed rational-box or rational-interval dense sequences. It provides unique regular PVMs, exact operator-domain and norm formulas, bounded Borel calculus, strongly continuous groups and precise generator domains under CC. PVM uniqueness supplies restriction and unitary transport, and later gives PVM intertwining from isometric group intertwining. These are the local spectral/group inputs used throughout the proof.

For this Coulomb operator, $r^{-1}{R}_0(z)$ is compact under CC. The free resolvent multiplier $m_z(p)=(\mathrm{\mid }p{\mathrm{\mid }}^2-z{)}^{-1}$ belongs to $L^2({\mathbb{R}}^3)$, so Fourier Cauchy–Schwarz gives $\mathrm{\parallel }{R}_0(z)f{\mathrm{\parallel }}_{\mathrm{\infty }}\le (2\pi {)}^{-3\mathrm{/}2}\mathrm{\parallel }{m}_z{\mathrm{\parallel }}_2\mathrm{\parallel }f{\mathrm{\parallel }}_2$. Its convolution kernel is $g_z=(2\pi {)}^{-3\mathrm{/}2}{\mathcal{F}}_p^{\ast }{m}_z\in L^2$. For bounded compactly supported $\chi$, the kernel $\chi (x)g_z(x-y)$ is in product $L^2$. Finite rational rectangular simple kernels approximate it there and give finite-rank operators; Cauchy–Schwarz bounds the operator norm by the kernel norm. Truncate $r^{-1}$ to $\epsilon <r<R$: the near-origin norm error is at most $\sqrt{4\pi \epsilon }(2\pi {)}^{-3\mathrm{/}2}\mathrm{\parallel }{m}_z{\mathrm{\parallel }}_2$, and the tail error at most $R^{-1}\mathrm{\mid }\mathrm{Im}z{\mathrm{\mid }}^{-1}$. Thus $r^{-1}{R}_0$ is a norm limit of compact operators. The resolvent identity makes $(C-z{)}^{-1}-R_0(z)=\gamma (C-z{)}^{-1}{r}^{-1}{R}_0(z)$ compact.

The essential-spectrum inference also has a specified countable construction. A real $\lambda$ is essential exactly when every spectral neighborhood has infinite rank. If it does, project the fixed dense sequence onto $E_A((\lambda -1\mathrm{/}n,\lambda +1\mathrm{/}n))H$ intersected with the orthogonal complement of previously constructed vectors, take the least nonzero projection and normalize. This yields an orthonormal sequence $u_n$ with $\mathrm{\parallel }(A-\lambda )u_n\mathrm{\parallel }\le 1\mathrm{/}n$, weakly null by the elementary finite Bessel inequality. Conversely, for a unit weakly null sequence with $(A-\lambda )u_n\to 0$, a finite-rank spectral neighborhood would give $E_{\epsilon }{u}_n\to 0$ and $\mathrm{\parallel }(I-E_{\epsilon })u_n\mathrm{\parallel }\le \mathrm{\parallel }(A-\lambda )u_n\mathrm{\parallel }\mathrm{/}\epsilon \to 0$, a contradiction. If $K=(C-z{)}^{-1}-(A-z{)}^{-1}$ is compact, $v_n=(C-z{)}^{-1}(A-z)u_n=u_n+K(A-z)u_n$ satisfies $v_n-u_n\to 0$ and $(C-\lambda )v_n=(A-\lambda )u_n+(z-\lambda )(v_n-u_n)\to 0$. Compact operators send bounded weakly null sequences to norm-null sequences: otherwise a subsequence with image norm bounded below has a norm-convergent image subsequence by compactness and CC, but its weak limit is zero, a contradiction. Normalize $v_n$ and reverse roles to get essential-spectrum equality. Explicit momentum shells give infinite-rank neighborhoods at every $\lambda \ge 0$ for $H_0$ and zero projections below zero, so ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s}}(C)=[0,\mathrm{\infty })$. Below zero finite-rank neighborhoods have finite-dimensional self-adjoint restrictions and finite sets of eigenvalues; every negative spectral point is isolated of finite multiplicity. Least rational isolating intervals make these points countable. The other negative points have zero-projection neighborhoods and a countable rational subcover, so the negative PVM is exhausted by the countable eigenprojections.

For each integer $l\ge 0$ let $V_l$ be the completion of $C_c^{\mathrm{\infty }}(0,\mathrm{\infty })$ in $\mathrm{\parallel }u{\mathrm{\parallel }}^2+\mathrm{\parallel }{u}^{\mathrm{\prime }}-u\mathrm{/}r{\mathrm{\parallel }}^2+l(l+1)\mathrm{\parallel }u\mathrm{/}r{\mathrm{\parallel }}^2$. Integration of $(\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{/}r{)}^{\mathrm{\prime }}$ gives $\mathrm{\parallel }{u}^{\mathrm{\prime }}-u\mathrm{/}r\mathrm{\parallel }=\mathrm{\parallel }{u}^{\mathrm{\prime }}\mathrm{\parallel }$ and $\mathrm{\parallel }u\mathrm{/}r\mathrm{\parallel }\le 2\mathrm{\parallel }{u}^{\mathrm{\prime }}\mathrm{\parallel }$, so $V_l=H_0^1(0,\mathrm{\infty })$ with equivalent norm for each fixed $l$. Closability is explicit: a core sequence tending to zero in $L^2$ with $u_n\mathrm{\prime }-u_n\mathrm{/}r$ convergent in $L^2$ has zero distributional limit on every compact subinterval, so that derivative limit is zero. This equality includes the endpoint: cutoffs vanishing below $\epsilon$ add a derivative error bounded in square by a constant times ${\int }_0^{2\epsilon }\mathrm{\mid }u\mathrm{/}r{\mathrm{\mid }}^2\to 0$; then cut off at infinity and smooth. The form $a_l[u,v]=\int (u^{\mathrm{\prime }}\overline{v^{\mathrm{\prime }}}+l(l+1)r^{-2}u\bar{v}-\gamma r^{-1}u\bar{v})$ is bounded on $V_l$, and $a_l[u,u]\ge \frac{1}{2}\mathrm{\parallel }{u}^{\mathrm{\prime }}{\mathrm{\parallel }}^2+l(l+1)\mathrm{\parallel }u\mathrm{/}r{\mathrm{\parallel }}^2-2{\gamma }^2\mathrm{\parallel }u{\mathrm{\parallel }}^2$. Thus $a_l+(2{\gamma }^2+1)⟨\cdot ,\cdot ⟩$ is a complete coercive inner product. Hilbert Riesz under CC solves its variational equation for every $L^2$ source. Define $A_{l}u=f$ by $a_l[u,v]=⟨f,v⟩$ for all $v\in V_l$. Compact smooth tests give dense domain, symmetry is immediate, and $A_l+2{\gamma }^2+1$ is onto. Solving this shifted equation for $(A_l^{\ast }+2{\gamma }^2+1)y$ shows that $y$ equals its solution, since the adjoint kernel is orthogonal to the onto range. Hence $A_l$ is self-adjoint. Its weak equation gives local $H^2$ regularity.

Use the CC polar unitary of [F3], $\mathcal{U}\psi =u_{lm}(r)=r\int \psi (r\omega )\overline{Y_{lm}}d\sigma$. On finite compact smooth radial sums its finite-block identity gives $\mathrm{\parallel }\psi {\mathrm{\parallel }}_{H^1}^2=\sum \mathrm{\parallel }{u}_{lm}{\mathrm{\parallel }}_{V_l}^2$ and $q(\psi ,\phi )=\sum a_l(u_{lm},v_{lm})$, with the genuine potential weight $r^{-1}$. These finite sums are $H^1$-dense: approximate by smooth functions, remove zero with radial cutoffs having gradient $O(1\mathrm{/}[r\log (1\mathrm{/}\epsilon )])$ and use Hardy to control the added derivative; compactly cut off and smooth away from zero. For a smooth annular $\psi$, its angular truncations ${\psi }_N=P_N\psi$ satisfy $\mathrm{\Delta }{\psi }_N=P_N\mathrm{\Delta }\psi$. Radial differentiation commutes with the projection; angular Green integration gives $⟨-{\mathrm{\Delta }}_{S^2}\psi (r\cdot ),Y_{lm}⟩=l(l+1)⟨\psi (r\cdot ),Y_{lm}⟩$. This uses the finite-span eigenidentity with a direct local Green calculation: excise polar caps, cancel periodic azimuth boundaries and let the cap radius tend to zero, since smooth intrinsic gradients are bounded and boundary lengths tend to zero. Parseval gives ${\psi }_N\to \psi$, $\mathrm{\Delta }{\psi }_N\to \mathrm{\Delta }\psi$ in $L^2$, and $\mathrm{\parallel }\mathrm{\nabla }(\psi -{\psi }_N){\mathrm{\parallel }}^2\le \mathrm{\parallel }\mathrm{\Delta }(\psi -{\psi }_N)\mathrm{\parallel }\mathrm{\parallel }\psi -{\psi }_N\mathrm{\parallel }\to 0$. Conversely finite core families are dense in the square-summable $V_l$ family space, so the finite isometry extends onto it from $H^1$. The full form identity extends by Cauchy–Schwarz; the Coulomb sum is uniformly bounded by $2(\sum \mathrm{\parallel }{u}_{lm}^{\mathrm{\prime }}{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}(\sum \mathrm{\parallel }{v}_{lm}{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}$. Define $\widehat{A}=⨁A_l$ on the square-summable graph domain. Its symmetry and coordinate resolvents with norm at most one at $\pm i$ prove self-adjointness. For $\psi \in H^2=D(C)$ and $\phi \in H^1$, weak integration by parts and Hardy give $q(\psi ,\phi )=⟨C\psi ,\phi ⟩$. Test $\phi ={\mathcal{U}}^{-1}(v{\delta }_{lm})$ with arbitrary $v\in V_l$: the exact form criterion gives $u_{lm}\in D(A_l)$ and $A_l{u}_{lm}=(\mathcal{U}C\psi {)}_{lm}$. The latter images are square summable; hence $C\subseteq {\mathcal{U}}^{\ast }\widehat{A}\mathcal{U}$. Both sides are self-adjoint, so equality follows by adjoints. This proves the exact block/domain identity under CC, also at $\gamma =0$, without borrowing the AC-bearing endpoint theorem.

For $k^2=z$ the regular radial solution is ${\phi }_l(k,r)=r^{l+1}{\sum }_{n\ge 0}{c}_n{r}^n$, with $c_0=1$ and $c_n=-(\gamma c_{n-1}+k^2{c}_{n-2})\mathrm{/}[n(n+2l+1)]$, $c_{-1}=0$. On compact $k$ sets choose $C\ge \max (1,\gamma ,\mathrm{\mid }k\mathrm{\mid })$; induction gives $\mathrm{\mid }{c}_n\mathrm{\mid }\le C^n\mathrm{/}n!$, since $[\gamma \mathrm{/}C+\mathrm{\mid }k{\mathrm{\mid }}^2(n-1)\mathrm{/}{C}^2]\mathrm{/}(n+2l+1)\le 1$. The series and its compact derivatives converge, giving a solution entire in $k$, real for real $k$, with regular coefficient one. Compact-interval continuation is given by the ordinary Volterra IVP series: for the first-order system its $n$th iterated integral is bounded by $M^n{L}^n\mathrm{/}n!$ uniformly on compact parameter sets, proving uniqueness, continuity, and holomorphic dependence rather than using a real-parameter theorem as a holomorphic premise.

For a compact set of $k$ in the closed first quadrant away from zero, set $a=\gamma \mathrm{/}(2k)$, $b=a^2+l(l+1)+ia$, $w(r)=e^{i(kr+a\log r)}$. On a common large tail $r\ge R$, the outgoing solution is $f=wv$, where $v(r)=1+b{\int }_r^{\mathrm{\infty }}[w(s{)}^2{\int }_r^{s}w(y{)}^{-2}dy]v(s)s^{-2}ds$. This is a single integral with the displayed oscillatory inner kernel. Indeed $\mathrm{\mid }w(s{)}^{2}w(y{)}^{-2}\mathrm{\mid }=e^{-2\mathrm{Im}k(s-y)}(s\mathrm{/}y{)}^{-2\mathrm{Im}a}\le 1$ for $s\ge y\ge R$ when $R\ge \gamma \mathrm{/}(2\mathrm{\mid }k{\mathrm{\mid }}^2)$. Put $h(y)=k+a\mathrm{/}y$ and choose $R$ also with $\mathrm{\mid }a\mathrm{\mid }\mathrm{/}R\le \mathrm{\mid }k\mathrm{\mid }\mathrm{/}2$. Integration by parts of $w(y{)}^{-2}$ using its derivative $-2ih(y)w(y{)}^{-2}$ bounds the inner kernel uniformly: its two endpoint terms are bounded by a constant times $\mathrm{\mid }k{\mathrm{\mid }}^{-1}$, and the remaining integral is bounded by a constant times ${\int }_r^s\mathrm{\mid }a\mathrm{\mid }{y}^{-2}\mathrm{/}\mathrm{\mid }k{\mathrm{\mid }}^{2}dy$. Thus the integral kernel is bounded by $C\mathrm{/}{s}^2$ uniformly on the compact parameter set. Its sup-norm operator has norm at most $C\mathrm{\mid }b\mathrm{\mid }\mathrm{/}R<1$ after increasing $R$, so the geometric Picard series constructs the unique bounded $v=1+O(r^{-1})$. Parameter continuity follows by uniformly dominated single-integral kernels; holomorphy follows by finite-interval holomorphy, uniform tail bounds, and uniform Picard convergence. Differentiation gives $v^{\mathrm{\prime }}=-bw(r{)}^{-2}{\int }_r^{\mathrm{\infty }}w(s{)}^{2}v(s)s^{-2}ds$ and $v^{\mathrm{\prime }\mathrm{\prime }}+2i(k+a\mathrm{/}r)v^{\mathrm{\prime }}-bv\mathrm{/}{r}^2=0$. Thus $f$ solves the exact equation and continues to finite $r>0$ by the IVP. For $\mathrm{Im}k>0$, $f,f^{\mathrm{\prime }}$ decay exponentially times powers; for real $k>0$, $v=1+O(r^{-1})$, $v^{\mathrm{\prime }}=O(r^{-1})$ give $W(\bar{f},f)=2ik$ by the limit at infinity.

Write $W_l(k)=W({\phi }_l,f)$ with $W(u,v)=u{v}^{\mathrm{\prime }}-u^{\mathrm{\prime }}v$. For real $k>0$ it is nonzero, since ${\phi }_l$ is nonzero and real and $W(\bar{f},f)=2ik\mathrm{\ne }0$. For nonreal $z$ a zero Wronskian would make the common regular-decaying solution belong to $V_l$ and satisfy the form eigenidentity, hence belong to $D(A_l)$ as a nonreal eigenvector, impossible. For $g$ supported in $[r_1,r_2]⋐(0,\mathrm{\infty })$ define $u(r)=-f(r){\int }_0^r{\phi }_l(s)g(s)\mathrm{/}{W}_{l}ds-{\phi }_l(r){\int }_r^{\mathrm{\infty }}f(s)g(s)\mathrm{/}{W}_{l}ds$. Below the source it is a multiple of ${\phi }_l$, above the source a multiple of $f$, so $u,u^{\mathrm{\prime }},u\mathrm{/}r\in L^2$ with zero trace, giving $u\in V_l$. The kernel $-{\phi }_l(r_{<})f(r_{>})\mathrm{/}{W}_l$ has derivative jump $-1$, and differentiation gives $(A_l-z)u=g$ weakly. Integration by parts against compact tests extends by form continuity to all $V_l$, proving the actual $A_l$ domain identity and thus $u=(A_l-z{)}^{-1}g$.

For real $k$, $\mathrm{Im}(-f\mathrm{/}{W}_l)=k{\phi }_l\mathrm{/}\mathrm{\mid }{W}_l{\mathrm{\mid }}^2$, so $\mathrm{Im}G=k{\phi }_l(r){\phi }_l(s)\mathrm{/}\mathrm{\mid }{W}_l{\mathrm{\mid }}^2$. The kernel is symmetric in $r,s$, so its local adjoint is its complex conjugate; the imaginary paired form for a complex source is therefore the integral of this imaginary kernel against $g(s)\overline{g(r)}$, giving the stated squared modulus. Continuity from the previous construction and the nonzero real Wronskian give uniform boundary-kernel convergence on compact source supports and compact positive energy intervals as $k=\sqrt{E+i\epsilon }\to \sqrt{E}$. Let ${\mu }_g(B)=⟨E_{A_l}(B)g,g⟩$. For $h\in C_c((0,\mathrm{\infty }))$, the local PVM calculus and Fubini give $(1\mathrm{/}\pi )\int h(E)\mathrm{Im}⟨(A_l-E-i\epsilon {)}^{-1}g,g⟩dE=\int (P_{\epsilon }\ast h)(\lambda )d{\mu }_g(\lambda )$ with $P_{\epsilon }(t)=\epsilon \mathrm{/}[\pi (t^2+{\epsilon }^2)]$. Its mass one and uniformly small tails imply $P_{\epsilon }\ast h\to h$ uniformly. Uniform boundary pairings give the other limit. Regular-measure uniqueness then proves $d{\mu }_g(E)={\rho }_l(E)\mathrm{\mid }\int {\phi }_l(\sqrt{E},r)g(r)dr{\mathrm{\mid }}^{2}dE$ on positive energies, with exactly ${\rho }_l(E)=\sqrt{E}\mathrm{/}[\pi \mathrm{\mid }{W}_l(\sqrt{E}){\mathrm{\mid }}^2]$. Compact-source approximation and $\mathrm{\mid }{\mu }_g(B)-{\mu }_{g_n}(B)\mathrm{\mid }\le (\mathrm{\parallel }g\mathrm{\parallel }+\mathrm{\parallel }{g}_n\mathrm{\parallel })\mathrm{\parallel }g-g_n\mathrm{\parallel }$, uniformly in $B$, extend positive absolute continuity to every vector.

Define $T_{l}g(E)=\sqrt{{\rho }_l(E)}\int {\phi }_l(\sqrt{E},r)g(r)dr$ on compact sources. Exhaust positive intervals to get $\mathrm{\parallel }{T}_{l}g{\mathrm{\parallel }}^2=⟨P_{+}g,g⟩$, where $P_{+}=E_{A_l}((0,\mathrm{\infty }))$. It extends as a contraction to all vectors, vanishes on $(I-P_{+})H$, and restricts to a closed-range isometry on $P_{+}H$. Polarization gives $T_l^{\ast }{M}_B{T}_l=E_{A_l}(B)$ for positive Borel sets. For the range projection $Q$, $Q{M}_B{T}_{l}g=T_l{E}_{A_l}(B)g$ and $\mathrm{\parallel }{M}_B{T}_{l}g{\mathrm{\parallel }}^2=⟨E_{A_l}(B)g,g⟩=\mathrm{\parallel }Q{M}_B{T}_{l}g{\mathrm{\parallel }}^2$; Pythagoras proves range invariance, so $Q$ commutes with every indicator.

A bounded operator $B$ on scalar $L^2(dE)$ commuting with all indicators is a multiplier: take $g_0(E)=e^{-E}$, $h=B{g}_0$, $b=h\mathrm{/}{g}_0$. The inequality $\mathrm{\parallel }{\mathbf{1}}_{D}h\mathrm{\parallel }\le \mathrm{\parallel }B\mathrm{\parallel }\mathrm{\parallel }{\mathbf{1}}_D{g}_0\mathrm{\parallel }$ proves $\mathrm{\mid }b\mathrm{\mid }\le \mathrm{\parallel }B\mathrm{\parallel }$ a.e. Simple approximation extends commutation to bounded measurable functions, so $B(u{g}_0)=ub{g}_0$; these vectors are dense by truncating energy and $\mathrm{\mid }f\mathrm{/}{g}_0\mathrm{\mid }$. Hence $B=M_b$. Completed-measure simple approximations provide a Borel version after countably many choices of Borel versions of level sets. Applied to the projection $Q$, this gives $Q=M_{{\mathbf{1}}_S}$. If $S^c$ had positive measure, the images of a fixed countable dense family of rational step functions on all intervals $[1\mathrm{/}j,j]$ would vanish there simultaneously off one null set. For each such energy, continuity of pairing on those compact intervals would force ${\phi }_l(\sqrt{E},\cdot )=0$ locally, contradicting its regular coefficient one. Thus $S$ is conull and ${\widehat{T}}_l$ is onto. Indicator intertwining, bounded simple approximation and energy truncation give the exact Borel correspondence and domain $\{h:Eh\in L^2\}$, with the original density normalization, for both $\gamma >0$ and $\gamma =0$.

At negative energy $-k^2$, $k>0$, substitute $u=r^{l+1}{e}^{-kr}v(2kr)$. The regular Kummer coefficients have $v_0=1$ and $v_{i+1}=(i-j)v_i\mathrm{/}[(i+1)(i+2l+2)]$ with $j=\gamma \mathrm{/}(2k)-l-1$. Their factorial majorant makes the series entire; it terminates exactly when $j\in {\mathbb{N}}_0$. A second solution obtained by reduction of order near zero behaves as $-r^{-l}\mathrm{/}(2l+1)$; it continues across nodes by the regular IVP. Its ratio $u\mathrm{/}r$ is unbounded. Any radial eigenvector transported into $D(C)=H^2$ has bounded $u\mathrm{/}r$ near zero by Fourier Cauchy–Schwarz ($⟨p{⟩}^{-2}\in L^2({\mathbb{R}}^3)$), so it is a multiple of the regular solution. In the nonterminating case its coefficients have fixed sign eventually, and their ratio times $i+1$ tends to one; for any $0<\epsilon <1\mathrm{/}2$ the tail therefore dominates $c(1-\epsilon {)}^i\mathrm{/}i!$ in absolute value. Thus $\mathrm{\mid }v(x)\mathrm{\mid }\ge c{e}^{(1-\epsilon )x}-P(x)$ for a polynomial $P$, and the radial $u$ is not $L^2$. In the terminating case the polynomial-exponential solution belongs to $V_l$ and its form eigenidentity puts it in $D(A_l)$. The negative radial levels are exactly $-{\gamma }^2\mathrm{/}(4n^2)$, $n\ge l+1$, each simple. The previously proved negative discrete-subspace closure exhausts the full negative subspace; at each $n$, the finite angular sum gives multiplicity ${\sum }_{l=0}^{n-1}(2l+1)=n^2$.

There is no zero eigenvector. Define $D_s\psi (x)=e^{3s\mathrm{/}2}\psi (e^{s}x)$, unitary on $L^2$. Weak derivative substitution gives ${\mathrm{\partial }}_j{D}_s=e^s{D}_s{\mathrm{\partial }}_j$ and the corresponding second-derivative identity, so $D_s{H}^2=H^2$. With $M\psi =\psi \mathrm{/}r$, change of variables gives $M{D}_s=e^s{D}_{s}M$, hence on exactly $H^2$, $C{D}_s\psi =e^{2s}{D}_{s}C\psi +\gamma (e^{2s}-e^s)D_{s}M\psi$. If $C\psi =0$, self-adjointness and $D_s\psi \in D(C)$ give $0=⟨C{D}_s\psi ,\psi ⟩=\gamma (e^{2s}-e^s)⟨D_{s}M\psi ,\psi ⟩$. For $\gamma >0$ and $s\mathrm{\ne }0$, divide by the nonzero scalar. Strong continuity of $D_s$ follows by dominated convergence first on compact smooth vectors and then by density and unitarity. Since $M\psi \in L^2$ by Hardy, letting $s\to 0$ gives $\int \mathrm{\mid }\psi {\mathrm{\mid }}^2\mathrm{/}r=0$, forcing $\psi =0$. This does not differentiate an unbounded dilation generator. At $\gamma =0$, the free Fourier multiplier has zero projection on $(-\mathrm{\infty },0]$, since $\{p=0\}$ is null. The countable radial resolution therefore has exactly the bound copies and the positive scalar continuum copies; in this representation $H_{\mathrm{a}\mathrm{c}}(C)=H_{\mathrm{c}}(C)=E_C((0,\mathrm{\infty }))H$.

For an annular smooth momentum profile $\psi$, $U_D(t)\psi$ belongs to $H^2$ and is continuously differentiable in the $H^2$ graph norm on each exterior half-line; Hardy makes the Coulomb map graph-continuous. The exact group-generator characterization proved above permits the product derivative $\frac{d}{dt}[e^{itC}{U}_D(t)\psi ]=-i{e}^{itC}[\gamma \mathrm{/}r-\gamma \mathrm{/}(2\mathrm{\mid }t\mathrm{\mid }\mathrm{\mid }P\mathrm{\mid })]U_D(t)\psi$. The remainder theorem bounds its norm by an integrable $C_{\psi }(1+\log \mathrm{\mid }t\mathrm{\mid })\mathrm{/}\mathrm{\mid }t{\mathrm{\mid }}^2$ for both signs. The Banach-valued fundamental theorem of calculus makes this vector Cauchy at each infinity. The approximants are unitary, so the limits preserve norm. Annular smooth profiles are dense by Fourier truncation and smoothing. For arbitrary $\psi$ use $\mathrm{\parallel }F(t)\psi -F(s)\psi \mathrm{\parallel }\le 2\mathrm{\parallel }\psi -{\psi }_n\mathrm{\parallel }+\mathrm{\parallel }F(t){\psi }_n-F(s){\psi }_n\mathrm{\parallel }$; both strong limits therefore exist on all $L^2$ as linear isometries.

For fixed real $s$, $e^{-isC}F(t)=e^{i(t-s)C}{U}_D(t)$. At large $\mathrm{\mid }t\mathrm{\mid }$ the signs of $t$ and $t-s$ agree, and $U_D(t)-U_D(t-s)e^{-is{H}_0}$ is a free unitary times the difference of momentum modifiers with phases $\mathrm{sign}(t)a(p)\log \mathrm{\mid }t\mathrm{\mid }$ and $\mathrm{sign}(t)a(p)\log \mathrm{\mid }t-s\mathrm{\mid }$. That difference tends pointwise to zero for $p\mathrm{\ne }0$ and is bounded by two; Fourier dominated convergence makes it strongly null. Consequently $e^{-isC}{\mathrm{\Omega }}_{D\pm }={\mathrm{\Omega }}_{D\pm }{e}^{-is{H}_0}$. Their closed ranges reduce the Coulomb group. Difference quotients give the restricted generator ${\mathrm{\Omega }}_{D\pm }{H}_0{\mathrm{\Omega }}_{D\pm }^{\ast }$ on the exact transported domain. The range projection commutes with the generator and its resolvents, hence reduces $C$. Uniqueness of the locally constructed PVM gives $E_C(B){\mathrm{\Omega }}_{D\pm }={\mathrm{\Omega }}_{D\pm }{E}_{H_0}(B)$. Since $E_{H_0}((-\mathrm{\infty },0])=0$, range inclusion in $E_C((0,\mathrm{\infty }))H$ is established before using the radial transforms.

The exact angular block decompositions above and the modifier as a bounded Borel function of $H_0$ make all comparison operators and their limits diagonal in each $(l,m)$ channel, with the same radial map for all $m$ at fixed $l$. Thus $B_{l,\pm }={\widehat{T}}_{\gamma ,l}{\mathrm{\Omega }}_{D\pm }^{(l)}{\widehat{T}}_{0,l}^{\ast }$ is an isometry on all scalar $L^2((0,\mathrm{\infty }),dE)$. Its spectral intertwining gives commutation with every energy indicator. The bounded-commutant construction above gives $B_{l,\pm }=M_{b_{l,\pm }}$ with a bounded Borel version. Isometry, tested on all finite-measure indicator functions, gives $\mathrm{\mid }{b}_{l,\pm }\mathrm{\mid }=1$ a.e.; multiplication by its complex conjugate is the bounded inverse, so each channel is onto its positive Coulomb space.

Countable orthogonal assembly gives $\mathrm{ran}{\mathrm{\Omega }}_{D\pm }=E_C((0,\mathrm{\infty }))H=H_{\mathrm{a}\mathrm{c}}(C)=H_{\mathrm{c}}(C)$. In detail, for a square-summable target channel family, apply the norm-one channel inverses; the source family remains square summable and its finite partial sums converge, and continuity of ${\mathrm{\Omega }}_{D\pm }$ gives that target as the image. The orthogonal bound-state projection remains the strong countable sum over $n\ge 1$ of the rank-$n^2$ projections at $-{\gamma }^2\mathrm{/}(4n^2)$. Both signs are unitaries from the free Hilbert space onto the same positive subspace; hence $S_D={\mathrm{\Omega }}_{D+}^{\ast }{\mathrm{\Omega }}_{D-}$ is unitary. This supplies all original channel and completeness conclusions without AC or a finite-bound-spectrum replacement.