# Scientific sources - Case47

## References

D. B. Milošević. 2013. *Reexamination of the improved strong-field approximation: Low-energy structures in the above-threshold-ionization spectra for short-range potentials*. *Physical Review A*, 88, 023417. [Source](https://doi.org/10.1103/PhysRevA.88.023417).

Complete energy/intermediate-channel first-Born ISFA for the supplied monochromatic short-range forward case. Complete/pole and coherent/direct consequences are required; selected-channel and PV subgroup figures are nonexclusive evidence examples, not extra obligations.

## Criterion-to-source crosswalk

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1 | Milošević (2013), article pp. 023417-3–5, Eqs. (20)–(22), (27)–(28), (42)–(44). Monochromatic field/order, forward target, p-state dipole and short-range Yukawa kernel, with common conventions. |
| F2 | Article pp. 023417-3–4, Eqs. (29)–(35), (37)–(41). Complete complex energy/channel amplitude, negative k angular measure and principal-value plus pole propagation; closed channels remain in off-shell amplitude. |
| F3, F4 | Article pp. 023417-3–5, Eqs. (28), (30), (32)–(36), (45); p. 023417-2, Eq. (8). Phase-compatible direct and pole-only comparator, differential rate and coherent addition. The energy-variable denominator changes the apparent Cauchy sign together with the kernel measure. |
| O1, O2, I1 | Article p. 023417-5, Fig. 2; p. 023417-6, Figs. 3–5 and principal-value discussion. Complete versus pole-only broad spectral difference; PV/channel component figures are canonical, not compulsory layouts or hidden peak targets. |
| O3, I3, M5 | Article p. 023417-7, Fig. 7 and direct/rescattered comparison paragraph; Eqs. (8), (28)–(29), (45). Coherent total can differ materially despite a smaller separate rescattering rate. Source 5D results support physics but do not establish local execution of that route. |
| O4, I2, M4 | Article p. 023417-6, Fig. 6 and selected n-prime=n paragraph. Bounded no-net-photon channel role in very-low-energy structure. Other channels/interference remain material; another discriminating grouping/removal analysis is equally acceptable. |
| M1, M2 | Article pp. 023417-3–4, Eqs. (29)–(41); p. 023417-6, Figs. 3–5 and amplitude-interference paragraph. On-shell retains the pole; complete propagation adds complex PV/off-shell content. Squared-sector rate addition is physically inequivalent. |
| M3 | Article pp. 023417-5–6, Figs. 2–6, closed/open-channel and forward-scattering discussion. Candidate-grounded intermediate-propagation mechanism; no mandatory closed/open or selected-channel panels. |
| M6 | Article pp. 023417-5–7, Figs. 2, 5, 7 and conclusion. Task-defined synthesis of complete/on-shell response and direct/rescattered coherence. A local selected-channel attribution gap is not another automatic hard gate on independently supported synthesis. |
| M7, V1 | Article pp. 023417-4–5, numerical target setting; pp. 023417-6–7, short-range/Coulomb and finite-pulse discussion. Established forward monochromatic first-Born scope; source 5D and UA are optional representations, not required claimed executions. |
| V2 | Article p. 023417-4, Eqs. (32)–(41); p. 023417-6, channel interference; p. 023417-7, Fig. 7. Task-defined triggered Cauchy, phase, channel, pole interpolation and coherent-normalization risks; no universal fixed quadrature or decomposition. |

## Formula and convention notes

$E_p=n\omega-I_p-U_p$, $x=k^2/2$, $x_{n\prime}=n\prime\omega-I_p-U_p$ and $T_n^R=\sum_{n\prime}\int_0^\infty f_{n\prime}(x)/(x-x_{n\prime}-i0)dx$ (Eqs. (31)–(32)). The kernel is $f_{n\prime}=-k\int d\Omega_k\,T_{pk}(n-n\prime)T_{ki}^{(0)}(n\prime)$, retaining the negative sign and the $k\,d\Omega_k$ energy measure (Eq. (33)); $T_{pk}$ includes the short-range scattering element and Bessel coefficient (Eq. (25)).

The required Cauchy identity is $(x-x_{n\prime}-i0)^{-1}=\mathcal P(x-x_{n\prime})^{-1}+i\pi\delta(x-x_{n\prime})$. Therefore $T_{n,\rm pole}^R=i\pi\sum_{x_{n\prime}\ge0}f_{n\prime}(x_{n\prime})$ (Eq. (36)); negative $x_{n\prime}$ channels have no pole but retain the complete off-shell integral. The alternative source $1/(y+i0)$ convention is equivalent only after transforming both denominator and kernel sign.

The direct Fourier coefficient is $T_{ki}^{(0)}(n)=(1/T)\int_0^Tdt\,\langle\mathbf k+\mathbf A(t)|\mathbf r\cdot\mathbf E(t)|\psi_i\rangle e^{i[\mathbf k\cdot\boldsymbol\alpha(t)+U_1(t)+n\omega t]}$ (Eqs. (27)–(28)), with $\dot{\boldsymbol\alpha}=\mathbf A$, $\dot U_1=A^2/2-U_p$ and $T=2\pi/\omega$. It must share the rescattering phase convention.

At fixed forward order, $w_n^R=2\pi p|T_n^R|^2$ and $w_n^{\rm coh}=2\pi p|T_n^D+T_n^R|^2$ (Eq. (45), amplitude identity Eq. (8)). The signed cross term is $4\pi p\operatorname{Re}(T_n^DT_n^{R*})$. Small $|T_n^R|^2$ alone does not bound this term or establish negligible rescattering.


## Scope of the source anchors

Source examples verify physical identity and robust structure; their numerical samples, plotting conventions and panel layout are not compulsory candidate deliverables. Reporting, reconstructibility, evidence attribution, conditional validation and Research Closure are examination-defined obligations, not additional results asserted by the article. A derivation or physical entailment is identified as such rather than presented as a directly printed source formula.