The Integral Over 2 Spherical Bessel Functions Multiplied by a Gaussian

Rami Mehrem

The Integral Over 2 Spherical Bessel Functions Multiplied by a Gaussian

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The Gaussian-weighted product of two spherical Bessel functions is exactly a finite sum of modified spherical Bessel functions under the angular-momentum triangle and parity conditions.

desk verdict A clean, narrowly scoped formula paper: the new finite-sum result is likely right, but the author should have numerically checked a generic case rather than leaning entirely on an earlier three-Bessel identity. read the letter →

arxiv 1908.07374 v2 pith:PJGOXKLZ submitted 2019-08-10 nucl-th

Rami Mehrem This is my paper · ORCID

classification nucl-th MSC 33C10

keywords sphericalBesselfunctionsmodifiedGaussianintegral3jsymbols6jclosurerelationnuclearscatteringpotential

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a closed-form, finite-sum evaluation of the 3j-weighted radial integral of two spherical Bessel functions times a Gaussian, an object that appears in nuclear scattering when harmonic-oscillator wavefunctions are used or when a Gaussian potential is evaluated in momentum space. The central claim is that the angular-momentum constraints carried by the 3j symbol, $|\lambda_2-\lambda_1| \le \lambda_3 \le \lambda_1+\lambda_2$ with $\lambda_1+\lambda_2+\lambda_3$ even, turn what would otherwise be an infinite hypergeometric series into a finite sum over modified spherical Bessel functions of the first kind. A sympathetic reader should take away that this class of oscillatory integrals has an exact, compact analytic representation, so slow and delicate numerical integration of rapidly oscillating Bessel integrands can be replaced by direct summation.

What carries the argument

The argument is carried by the spherical Bessel closure relation, $$\int_0^\infty $k^{2}$ j_L(kr)j_L(kr')\,dk=\frac{\pi}{$2r^{2}$}\delta(r-r'),$$ which inserts a third spherical Bessel function $j_{\lambda_3}(k_3r)$ into the integrand and converts the target integral into an integral over $k_3$ of a known three-spherical-Bessel integral (equation 2.1) times a one-dimensional Gaussian transform (equation 2.5). The change of variable $k_3\to \Delta=(k_1^2+k_2^2-k_3^2)/(2k_1k_2)$ restricts $\Delta$ to $[-1,1]$ and brings in the generating expansion $$\exp\!\left(\frac{k_1k_2\$\Delta$}{2\$\alpha$}\right)=\sum_L (2L+1)P_L(\$\Delta$)\,i_L\!\left(\frac{k_1k_2}{2\$\alpha$}\right),$$ where $i_L(x)=\sqrt{\pi/(2x)}\,I_{L+1/2}(x)$ is the modified spherical Bessel function of the first kind. This expansion is what turns the leftover integral into the finite sum of modified spherical Bessel functions.

What would settle it

Take an admissible nontrivial case, for example $\alpha=1$, $k_1=1$, $k_2=2$, and $\lambda_1=\lambda_2=\lambda_3=2$, evaluate the right-hand side of (2.9), and compare it with high-precision numerical quadrature of the original oscillatory integral over $r$. Agreement to machine precision would support the identity; any disagreement beyond the quadrature error would refute it. Applying the same numerical check to the quoted three-Bessel formula (2.1) would then show whether the trouble lies in the starting identity or in the later reduction.

Watch

Extended reading notes

Core claim

The paper's central discovery is the closed identity (2.9): for nonnegative integers $\lambda_1,\lambda_2,\lambda_3$ with $|\lambda_2-\lambda_1|\le \lambda_3\le \lambda_1+\lambda_2$ and $\lambda_1+\lambda_2+\lambda_3$ even, and for positive $k_1,k_2,\alpha$, the 3j-weighted integral $$\begin{pmatrix}\lambda_1 & \lambda_2 & \lambda_3\\ 0 & 0 & 0\end{pmatrix}\int_0^\infty $r^{{\lambda_3+2}}$$e^{{-\alpha r^2}}$ j_{\lambda_1}(k_1r)j_{\lambda_2}(k_2r)\,dr$$ equals a finite sum over $L$ from $0$ to $\lambda_3$ and over the finite range of $l$ allowed by the angular-momentum coefficients, with terms made of 3j symbols, 6j symbols, and modified spherical Bessel functions $i_l(k_1k_2/(2\alpha))$, multiplied by the Gaussian factor $\exp[-(k_1^2+k_2^2)/(4\alpha)]$. The finiteness is the point: the 3j constraints truncate sums that would otherwise run forever as an infinite hypergeometric series. Setting $\lambda_3=0$, which forces $\lambda_1=\lambda_2=\lambda$, reduces the identity to the single-term formula with $i_\lambda(k_1k_2/(2\alpha))$, recovering the previously published special case.

Load-bearing premise

The derivation rests on the three-spherical-Bessel integral (2.1), quoted from the author's earlier papers without being re-derived or independently checked here; if that quoted identity has a hidden restriction or an error, the final closed form (2.9) would not stand.

Editorial extensions

If this is right

The 3j-weighted two-spherical-Bessel Gaussian integral can be evaluated exactly by summing finitely many 3j and 6j coefficients and modified spherical Bessel functions; no infinite series or oscillatory radial quadrature is needed.
The special case $\lambda_3=0$ (hence $\lambda_1=\lambda_2$) reduces the general result to a known one-term formula, providing an internal consistency check.
Nuclear scattering calculations using harmonic-oscillator target wavefunctions and momentum-space Gaussian potentials can substitute this closed form for numerical integration of the radial integral.
The derivation shows a general route: inserting the spherical Bessel closure relation reduces a product of two Bessel functions to a known three-Bessel integral whose angular-momentum structure is already understood.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same closure-relation insertion should produce finite closed forms for Gaussian-weighted products of three or more spherical Bessel functions whenever angular-momentum constraints truncate the intermediate sums; the paper does not carry out such cases.
Because the 3j symbol vanishes outside the stated triangle and parity window, identity (2.9) does not directly give the unweighted integral; using the orthogonality of 3j symbols to unwind the weight would yield the bare integral for arbitrary integer $\lambda_3$.
The right-hand side of (2.9) is analytic in $\alpha$ on the half-plane $\mathrm{Re}\,\alpha>0$, so the identity plausibly extends by analytic continuation from real positive $\alpha$ to complex Gaussian parameters, covering damped or oscillating momentum-space potentials.
The finite-sum form is well suited to reuse in momentum-space codes that currently evaluate such integrals by adaptive quadrature, since it replaces the expensive oscillatory integration with a small number of spherical Bessel evaluations.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main formula (2.9) is genuinely new: a finite sum over modified spherical Bessel functions for an integral that Gradshteyn and Ryzhik only gives as an infinite series. The derivation is transparent — closure relation, one-Bessel Gaussian integral, the author's earlier three-Bessel formula (2.1), and a Legendre expansion. The λ3=0 limit reproduces Tabakin–Davies, which is a solid independent check.

The load-bearing piece is (2.1), imported from earlier papers and not re-derived here. That is a real gap in self-containedness. The check in (2.10) only exercises λ3=0, which collapses the sums and does not stress the 3j/6j structure, the powers of k1 and k2, or the finite-sum combinatorics for generic triples. A numerical spot-check for one or two nonzero λ3 would have settled this. The closure relation (2.3) is written with an asymmetric 1/r² factor, but that is harmless under the integrals. I checked the algebra of the Δ to k3 change of variable and the Legendre expansion; the factors are correct.

Nothing here is a fatal flaw. The citation of the author's own prior work is appropriate, since (2.1) is published in J. Phys. A and Phys. Rev. C. The paper is exactly what it appears to be: a useful addition to the toolbox, not a new mechanism or a resolution of a big question. The target audience is people doing nuclear scattering or momentum-space potentials with Gaussian forms.

It deserves a serious referee. My recommendation would be to accept after the author adds a short numerical verification for a generic triple and cleans up the typos. The absence of that verification does not make the result suspect, but it would make the paper stronger.

Referee Report

0 major / 4 minor

Summary. This paper evaluates the integral of two spherical Bessel functions multiplied by a Gaussian and a power factor, with a 3j symbol prefactor, obtaining the finite-sum expression in Eq. (2.9). The derivation uses the closure relation for spherical Bessel functions, the known one-Bessel Gaussian integral (Eq. 2.5), a three-spherical-Bessel integral taken from the author's prior work (Eq. 2.1), and the Legendre expansion of the exponential (Eq. 2.8). The λ3=0 limit reproduces a known result of Tabakin and Davies (Eq. 2.10).

Significance. If correct, the finite-sum formula is a useful analytic tool for nuclear scattering and momentum-space potential calculations, avoiding the infinite series of the standard Gradshteyn–Ryzhik expression. The derivation is compact and logically sound, and the λ3=0 check provides an independent consistency test. The paper ships no numerical verification for generic triples, which is the main residual risk, but the central mathematical structure appears sound.

minor comments (4)

[§2, Eq. (2.1)] The central result Eq. (2.9) relies on the three-spherical-Bessel identity Eq. (2.1), which is cited from refs. [8,10,12] but not re-derived or numerically checked in this manuscript. The λ3=0 limit only exercises the single-term L=0 contribution and does not test the 3j/6j recoupling structure, the (k2/k1)^L factors, or the finite-sum combinatorics. A numeric spot-check for a generic triple (e.g., λ1=2, λ2=3, λ3=1) would substantially increase confidence in the final formula.
[§2, Eqs. (2.5), (2.9)] The typesetting of the square roots is ambiguous: Eq. (2.5) appears to read sqrt(π)/2 but must be sqrt(π/2) to be consistent with the known λ=0 integral and with Eq. (2.10). Similarly, the prefactor in Eq. (2.9) should read sqrt(π/2), not sqrt(π)/2. Please ensure the radical sign covers the full fraction.
[Title and text] There are several typos: 'MUL TIPLIED' in the title, 'Refrences' at the start of Section 2, and 'applicabale' before Eq. (2.9). These should be corrected.
[§2, Eq. (2.8)] The Legendre expansion of the exponential in modified spherical Bessel functions is standard, but a specific reference (e.g., Abramowitz and Stegun) would help readers unfamiliar with this identity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main formula is derived from standard identities plus an independently published triple-spherical-Bessel result, not from the quantity it claims to evaluate.

full rationale

The derivation of (2.9) uses the closure relation (2.3), the standard Gaussian-spherical-Bessel integral (2.5) from Gradshteyn-Ryzhik, the Rayleigh-type expansion (2.8), and the published triple-spherical-Bessel integral (2.1) from refs. [8,10,12]. Eq. (2.1) is a parameter-free formula for a different integral (three spherical Bessel functions, no Gaussian) and its stated assumptions do not include the target integral; it is therefore independent support rather than an input defined in terms of the result. The final formula is benchmarked at lambda3=0 against Tabakin and Davies [19] in (2.10). No fitted parameter is renamed as a prediction, and no equation is constructed to be equivalent to the target integral by definition. The fact that (2.1) is drawn partly from the author's prior work is a normal citation of a published lemma; it would be a verification concern if that lemma were unproven, but it is not circularity in the derivation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The derivation relies on standard identities (closure relation, Legendre expansion, G&R integral) and on a specialized three-Bessel integral from the author's prior work. No free parameters or invented entities are introduced.

assumptions (5)

standard math Closure relation for spherical Bessel functions: ∫ k² j_L(kr) j_L(kr') dk = π/(2r²) δ(r-r') (eq. 2.3).
Used in eq. (2.4) to insert a complete set of intermediate Bessel states; standard identity in the theory of spherical Bessel functions.
standard math Single-Bessel Gaussian integral (eq. 2.5), quoted from Gradshteyn-Ryzhik 6.633.
Provides the closed form for ∫ r'^{λ3+2} e^{-αr'^2} jλ3(k3r') dr' used in eq. (2.4).
domain assumption Three-spherical-Bessel integral formula (eq. 2.1) from Mehrem, Londergan, Macfarlane (1991) and related papers.
This is the main input to the derivation; it is a specialized published result, largely from the author's own prior work, and is not re-derived or numerically checked in the present paper.
standard math Legendre expansion of exp(zΔ) in modified spherical Bessel functions (eq. 2.8).
Used to evaluate the angular integral in eq. (2.7); standard generating function identity.
domain assumption The 3j symbol conditions (triangle inequality and even parity) on λ1, λ2, λ3.
These conditions guarantee the three-Bessel integral and the finite sums are well defined; they are standard angular momentum selection rules.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The Integral Over 2 Spherical Bessel Functions Multiplied by a Gaussian." pith.science (2026). https://pith.science/paper/PJGOXKLZ

@misc{pith2026190807374,
  author       = {Pith},
  title        = {Pith review of: The Integral Over 2 Spherical Bessel Functions Multiplied by a Gaussian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PJGOXKLZ}},
  note         = {Machine review of arXiv:1908.07374}
}

abstract

In this paper, the integral $\pmatrix{\lambda_1 &\lambda_2 &\lambda_3\cr 0 &0 &0\cr}\, \int_0^\infty \, r^{\lambda_3+2}\, \exp{(-\alpha r^2)}\, j_{\lambda_1}(k_1r) \,j_{\lambda_2}(k_2r) \,dr$, where $k_1$, $k_2$ and $\alpha$ are positive, is evaluated analytically. The result is a finite sum over the modified spherical Bessel function of the first kind. This result will be useful for nuclear scattering calculations, where harmonic oscillator nuclear wavefunctions are used or when evaluating momentum space matrix elements for a Gaussian potential.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Large-scale Modeling of the Observed Power Spectrum Multipoles
astro-ph.CO 2026-01 conditional novelty 6.0 of 10

Eq. (25) computes Yamamoto power-spectrum multipoles in linear theory as weighted sums of discrete spherical Fourier-Bessel power-spectrum modes, unifying wide-angle, redshift-evolution, window, and integral-constrain...

Reference graph

Works this paper leans on

23 extracted references · 21 canonical work pages · cited by 1 Pith paper

[1]

G. N. Watson, A Treatise on the Theory of Bessel Functions , (Cambridge Univ. Press, 1944)

work page 1944
[2]

Sawaguri and W

T. Sawaguri and W. Tobocman, Journ. Math. Phys. 8, 2223 (1967)

work page 1967
[3]

A. D. Jackson and L. Maximon, SIAM J. Math. Anal. 3, 446 (1972)

work page 1972
[4]

Anni and L

R. Anni and L. Taﬀara, Nuovo Cimento 22A, 12 (1974)

work page 1974
[5]

Elbaz, J

E. Elbaz, J. Meyer and R. Nahabetian, Lett. Nuovo Cimento 10, 418 (1974)

work page 1974
[6]

Elbaz, Riv

E. Elbaz, Riv. Nuovo Cimento 5, 561 (1975)

work page 1975
[7]

Gervais and H

A. Gervais and H. Navelet, J. Math. Phys. 26, 633, 645 (1985)

work page 1985
[8]

Mehrem, J.T

R. Mehrem, J.T. Londergan and M. H. Macfarlane, J. Phys. A 24, 1435 (1991)

work page 1991

Show all 23 references

[9]

Mehrem, Ph.D

R. Mehrem, Ph.D. Thesis , (Indiana University 1992)(unpublished)

1992
[10]

Mehrem, J.T

R. Mehrem, J.T. Londergan and G. E. Walker, Phys. Rev. C 48, 1192 (1993)

1993
[11]

Chen and J

J. Chen and J. Su, Phys. Rev. D 69, 076003 (2004)

2004
[12]

Mehrem and A

R. Mehrem and A. Hohenegger, J. Phys. A 43, 455204 (2010), arXiv: math- ph/1006.2108, 2010

2010 arXiv
[13]

Mehrem, Appl

R. Mehrem, Appl. Math. Comp. 217, 5360 (2011), arXiv: math-ph/0909.0494, 2010. 6

2011 arXiv
[14]

Y.L.Luke, Integrals of Bessel Functions , (Dover Publications, 2014)

2014
[15]

Mathews and E

G.B. Mathews and E. Meissel, A treatise on Bessel Functions and Their Appli- cations to Physics , (Wentworth Press, 2019)

2019
[16]

K. T. R. Davies, M. R. Strayer and G. D. White, J. Phys. G14, 961 (1988)

1988
[17]

K. T. R. Davies, J. Phys. G14, 973 (1988)

1988
[18]

Van Deun and R

J. Van Deun and R. Cools, ACM Trans. Math. Softw. 32, 580 (2006)

2006
[19]

Tabakin and K.T.R

F. Tabakin and K.T.R. Davies, Phys. Rev., 150, 793 (1966)

1966
[20]

G. D. White, K. T. R. Davies and P. J. Siemens, Ann. Phys. (N Y) 187, 198 (1988)

1988
[21]

Gradshteyn and I.M

I.S. Gradshteyn and I.M. Ryzhik: Table of Integrals, Series and Products (Aca- demic Press, San Diego, CA, 7th edition, 2007)

2007
[22]

Brink and G.R

D.M. Brink and G.R. Satchler, Angular Momentum (Oxford University Press, London 1962)

1962
[23]

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univer- sity Press, Princeton, 1957). 7

1957

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

G. N. Watson, A Treatise on the Theory of Bessel Functions , (Cambridge Univ. Press, 1944)

work page 1944

[2] [2]

Sawaguri and W

T. Sawaguri and W. Tobocman, Journ. Math. Phys. 8, 2223 (1967)

work page 1967

[3] [3]

A. D. Jackson and L. Maximon, SIAM J. Math. Anal. 3, 446 (1972)

work page 1972

[4] [4]

Anni and L

R. Anni and L. Taﬀara, Nuovo Cimento 22A, 12 (1974)

work page 1974

[5] [5]

Elbaz, J

E. Elbaz, J. Meyer and R. Nahabetian, Lett. Nuovo Cimento 10, 418 (1974)

work page 1974

[6] [6]

Elbaz, Riv

E. Elbaz, Riv. Nuovo Cimento 5, 561 (1975)

work page 1975

[7] [7]

Gervais and H

A. Gervais and H. Navelet, J. Math. Phys. 26, 633, 645 (1985)

work page 1985

[8] [8]

Mehrem, J.T

R. Mehrem, J.T. Londergan and M. H. Macfarlane, J. Phys. A 24, 1435 (1991)

work page 1991