A complete Coulomb forcing measure

With $\psi=u(r)Y_{10}(\widehat r)/r$, the radial forcing and Friedrichs operator are

Integration of $r^4e^{-2r}$ gives $\|f\|^2=1$. Direct differentiation gives $h_1u_f=(-1/2+1/r)u_f$, whence $\langle f,hf\rangle=0$.

We specify the continuum normalization because an energy cross-section convention could otherwise introduce a wrong Jacobian. The regular Coulomb function is normalized by its unit-amplitude large-radius oscillation. In the standard convention [10],

For $H_1^+=G_1+iF_1$, the Wronskian in $r$ is $W_r(F_1,H_1^+)=-k$. The upper-boundary radial resolvent kernel is

Its derivative jump is $-2$, as required by the coefficient $-1/2$ in $h_1$. Its imaginary part is $(2/k)F_1(r)F_1(r')$. Stone's formula therefore gives the measure $u_ku_k\,dk$, or $u_ku_k/k$ per unit energy $E=k^2/2$. The Coulomb asymptotics used to choose $H_1^+$ are asymptotic statements at large radius, not exact finite-radius equalities.

Here is the overlap calculation. Put $a=2+i/k$, $s=1+ik$, $z=2ik/(1+ik)$. Laplace integration of the confluent hypergeometric series, first in its absolute-convergence region and then by analytic continuation, gives

The coefficient identity $(5)_j/(4)_j=1+j/4$ implies

The last bracket is $1/[2(1-ik)]$. Consequently the integral is

The continuous logarithm of the ratio is $-2i\arctan k$. Its contribution to the squared modulus is $e^{-4\arctan(k)/k}$. Multiplication by the normalized radial prefactors gives $|\langle u_k,u_f\rangle|^2=k\rho_C(k^2/2)$. The factor $1/k$ from Stone's energy measure proves (9.4).

The normalized negative-energy eigenfunction in this channel is

Insert $L_j^3=(4)_j\,{}_1F_1(-j;4;\cdot)/j!$ into the same Laplace identity. The bracket is now $n/[2(n-1)]$ and gives (9.3); in particular $d_2^2=32768/59049$.

For completeness, the radial regular/decaying solutions have only the simple negative poles $-1/(2n^2)$, $n\geq2$. Their residues give precisely these normalized eigenprojections. On every compact positive-energy interval, (9.7) has continuous boundary values against the exponentially decaying forcing, so its forcing measure there is absolutely continuous. A remaining singular measure could only be supported at zero. The regular zero-energy solution is proportional to $\sqrt r\,J_3(\sqrt{8r})$; its large-radius oscillatory magnitude is of order $r^{1/4}$ and it is not $L^2$. Thus zero is not an eigenvalue and supplies no atom. A singular continuous measure cannot be supported on a single point. This proves completeness of (9.2). The already evaluated norm and first energy expectation now give (9.5); numerical quadrature is not used to establish them.

To prove the whole-continuum ceiling, let $A(k)=\arctan k-k/(1+k^2)$. Since

we have $A(k)\leq2k^3/[3(1+k^2)]$. The logarithmic derivative of $(1+k^2/2)\rho_C(k^2/2)$ is consequently at most

For $k\leq1$, use $e^x-1\geq x^3/6$ to bound the positive term by $3k/(2\pi^2)$; this leaves a strictly negative margin. For $k\geq1$, use $e^x-1>x$ to bound it by $1/k$, whereas the negative magnitude is at least $8/(3k)$. The threshold limit is $\rho_*$. This proves the weighted ceiling and hence the unweighted one.

In the cyclic spectral representation generated by $f$, the continuum wave equals

Extend the time input by zero outside $[0,t]$. Hilbert-valued Plancherel, integrated first on the whole frequency line, gives the factor $2\pi$ for this unnormalized Fourier integral. Restricting to $E>0$ and using (9.6) gives both estimates in (9.9). The source form commutes its own propagation, so the same argument applied to $H_s^{1/2}a$ gives (9.10).