SphericalBessel Namespace Reference

Detailed Description

Spherical Bessel functions j_L(x) and y_L(x), and related utilities.

Provides an approximate version (good to ~1e-9) and exact GSL-backed versions, plus cell-averaged vectors for integration on coarse grids, and a lookup table.

Classes
class	JL_table
	Lookup table of spherical Bessel functions: j_L(q*r) = J[L][q][r]. More...

Functions
double	jL (int L, double x)
	Spherical Bessel function j_L(x); uses low-x approximation where appropriate.
double	exactGSL_JL (int L, double x)
	Spherical Bessel function of first kind via GSL.
double	exactGSL_YL (int L, double x)
	Spherical Bessel functions of second kind.
double	yL (int L, double x)
	Spherical Bessel function of second kind, y_L(x)
double	PhiL (int L, double x, bool tilde=false)
	Phi_L(x) = [(2L+1)!! / x^L] * j_L(x). If tilde=true, returns 1 - Phi_L(x)
double	PsiL (int L, double x, bool tilde=false)
	Psi_L(x) = [x^{L+1} / (2L-1)!!] * y_L(x). If tilde=true, returns 1 - Psi_L(x)
std::vector< double >	fillBesselVec (int l, const std::vector< double > &xvec, bool cell_average=true)
	Fills a vector with the spherical Bessel function j_L sampled on the supplied grid; oscillation-resolving cell-average is used where needed.
std::vector< double >	fillBesselVec_kr (int l, double k, const std::vector< double > &rvec, bool cell_average=true)
	As fillBesselVec, with the argument of j_L being k*r instead of x.
void	fillBesselVec_kr (int l, double k, const std::vector< double > &r, std::vector< double > *jl, bool cell_average=true)
	In-place variant of fillBesselVec_kr; writes into a caller-owned vector.

Function Documentation

jL()

double SphericalBessel::jL	(	int	L,
		double	x
	)

Spherical Bessel function j_L(x); uses low-x approximation where appropriate.

exactGSL_JL()

double SphericalBessel::exactGSL_JL	(	int	L,
		double	x
	)

Spherical Bessel function of first kind via GSL.

exactGSL_YL()

double SphericalBessel::exactGSL_YL	(	int	L,
		double	x
	)

Spherical Bessel functions of second kind.

yL()

double SphericalBessel::yL	(	int	L,
		double	x
	)

Spherical Bessel function of second kind, y_L(x)

PhiL()

double SphericalBessel::PhiL	(	int	L,
		double	x,
		bool	tilde
	)

Phi_L(x) = [(2L+1)!! / x^L] * j_L(x). If tilde=true, returns 1 - Phi_L(x)

PsiL()

double SphericalBessel::PsiL	(	int	L,
		double	x,
		bool	tilde
	)

Psi_L(x) = [x^{L+1} / (2L-1)!!] * y_L(x). If tilde=true, returns 1 - Psi_L(x)

fillBesselVec()

std::vector< double > SphericalBessel::fillBesselVec	(	int	l,
		const std::vector< double > &	xvec,
		bool	cell_average = true
	)

Fills a vector with the spherical Bessel function j_L sampled on the supplied grid; oscillation-resolving cell-average is used where needed.

Returns a vector v such that v[i] is the value (or cell-average) of j_L on the grid point xvec[i]. The "cell" at index i is the interval bounded by the midpoints to its neighbours (asymmetric for the first/last point).

This vector is intended for integration of the form \( \int j_L(x)\,\psi(x)\,dx \approx \sum_i w_i\,v_i\,\psi(x_i) \) against a smooth wavefunction \(\psi\) sampled on the same grid. When the grid cell at index i resolves the oscillation of j_L ( \(\Delta x_i \le \lambda_j / N\) with \(\lambda_j = 2\pi\) asymptotically and N = 10), v[i] is the point value j_L(xvec[i]). When the cell is too coarse, v[i] is the cell average

\[ v_i = \frac{1}{\Delta x_i}\int_{\text{cell}_i} j_L(x)\,dx , \]

computed by trapezoidal sub-sampling with enough sub-points to resolve j_L on the cell. Using these cell averages in the Riemann-sum quadrature is equivalent to a Filon-style integration: the (smooth) factor \(\psi\) is treated as piecewise-constant per cell while the oscillation of j_L is integrated exactly within each cell. This recovers the correct Riemann-Lebesgue (~1/x) suppression at large x and removes the aliasing seen with pure point-sampling on a coarse grid.

Parameters

l	Order of the spherical Bessel function.
xvec	Grid (monotonic) on which j_L is sampled.
cell_average	If true (default), enables the cell averaging described above. If false, every entry is a pure point sample j_L(xvec[i]); use this for inspection / plotting or to reproduce legacy behaviour.

Returns

Vector v with v[i] = j_L(xvec[i]) or its cell average; same size as xvec.

Warning

The returned values are NOT pure point samples on cells where averaging triggers. They are only meaningful in conjunction with a Riemann-sum-style quadrature using the same grid weights.

fillBesselVec_kr() [1/2]

std::vector< double > SphericalBessel::fillBesselVec_kr	(	int	l,
		double	k,
		const std::vector< double > &	rvec,
		bool	cell_average = true
	)

As fillBesselVec, with the argument of j_L being k*r instead of x.

Returns a vector v with v[i] = j_L(k*rvec[i]) or, where the cell at index i is too coarse for the oscillation of j_L(kr) (cell width \(\Delta r_i > 2\pi / (k\,N)\) with N = 10), the cell average

\[ v_i = \frac{1}{\Delta r_i}\int_{\text{cell}_i} j_L(k r)\,dr . \]

See fillBesselVec for the usage and warning.

Parameters

l	Order of the spherical Bessel function.
k	Momentum (or other) factor multiplying r in the argument.
rvec	Radial grid on which j_L(kr) is sampled.
cell_average	If true (default), enables cell averaging; if false, point-samples j_L(k*rvec[i]) at every grid point.

Returns

Vector v of same length as rvec.

fillBesselVec_kr() [2/2]

void SphericalBessel::fillBesselVec_kr	(	int	l,
		double	k,
		const std::vector< double > &	r,
		std::vector< double > *	jl,
		bool	cell_average = true
	)

In-place variant of fillBesselVec_kr; writes into a caller-owned vector.

Identical semantics to the returning overload (including the cell-averaging behaviour); jl is resized to r.size().

Parameters

l	Order of the spherical Bessel function.
k	Momentum factor multiplying r in the argument.
r	Radial grid.
jl	Output vector (resized internally); must be non-null.
cell_average	If true (default), enables cell averaging; if false, point-samples j_L(k*r[i]) at every grid point.