G. The Bessel equation in spherical coordinates

When solving the equation for the geometry in spherical coordinates, the following equation for the radius arises

r2⋅d2R(r)dr2\displaystyle r^2 \cdot \frac{\displaystyle d^2 R(r)}{\displaystyle dr^2} +2⋅r⋅dR(r)dr\displaystyle + 2 \cdot r \cdot \frac{\displaystyle d R(r)}{\displaystyle dr} +(r2⋅γ2−k⋅(k+1))⋅R(r)\displaystyle + \left( r^2 \cdot \gamma^2 - k \cdot (k + 1) \right) \cdot R(r) =0,\displaystyle = 0,
(G.1)

where γ\gamma is the eigenvalue, kk is a nonnegative integer.

Let us make the substitution R(r)R(r) =1r⋅R^(r)= \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot \widehat{R}(r) and carry out the transformations

dR(r)dr\displaystyle \frac{\displaystyle d R(r)}{\displaystyle dr} =ddr(1r⋅R^(r))\displaystyle = \frac{\displaystyle d}{\displaystyle dr} \left( \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot \widehat{R}(r) \right) =−12⋅r⋅r⋅R^(r)\displaystyle = - \frac{\displaystyle 1}{\displaystyle 2 \cdot r \cdot \sqrt r} \cdot \widehat{R}(r) +1r⋅dR^(r)dr,\displaystyle + \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot \frac{\displaystyle d \widehat{R}(r)}{\displaystyle dr},
d2R(r)dr2\displaystyle \frac{\displaystyle d^2 R(r)}{\displaystyle dr^2} =−12⋅ddr(1r⋅r⋅R^(r))\displaystyle = - \frac{\displaystyle 1}{\displaystyle 2} \cdot \frac{\displaystyle d}{\displaystyle dr} \left( \frac{\displaystyle 1}{\displaystyle r \cdot \sqrt r} \cdot \widehat{R}(r) \right) +ddr(1r⋅dR^(r)dr),\displaystyle + \frac{\displaystyle d}{\displaystyle dr} \left( \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot \frac{\displaystyle d \widehat{R}(r)}{\displaystyle dr} \right),
−12⋅ddr(1r⋅r⋅R^(r))\displaystyle - \frac{\displaystyle 1}{\displaystyle 2} \cdot \frac{\displaystyle d}{\displaystyle dr} \left( \frac{\displaystyle 1}{\displaystyle r \cdot \sqrt r} \cdot \widehat{R}(r) \right) =34⋅1r2⋅r⋅R^(r)\displaystyle = \frac{\displaystyle 3}{\displaystyle 4} \cdot \frac{\displaystyle 1}{\displaystyle r^2 \cdot \sqrt r} \cdot \widehat{R}(r) −12⋅1r⋅r⋅dR^(r)dr,\displaystyle - \frac{\displaystyle 1}{\displaystyle 2} \cdot \frac{\displaystyle 1}{\displaystyle r \cdot \sqrt r} \cdot \frac{\displaystyle d \widehat{R}(r)}{\displaystyle dr},
ddr(1r⋅dR^(r)dr)\displaystyle \frac{\displaystyle d}{\displaystyle dr} \left( \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot \frac{\displaystyle d \widehat{R}(r)}{\displaystyle dr} \right) =−12⋅r⋅r⋅dR^(r)dr\displaystyle = - \frac{\displaystyle 1}{\displaystyle 2 \cdot r \cdot \sqrt r} \cdot \frac{\displaystyle d \widehat{R}(r)}{\displaystyle dr} +1r⋅d2R^(r)dr2,\displaystyle + \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot \frac{\displaystyle d^2 \widehat{R}(r)}{\displaystyle dr^2},
d2R(r)dr2\displaystyle \frac{\displaystyle d^2 R(r)}{\displaystyle dr^2} =1r⋅d2R^(r)dr2\displaystyle = \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot \frac{\displaystyle d^2 \widehat{R}(r)}{\displaystyle dr^2} −1r⋅r⋅dR^(r)dr\displaystyle - \frac{\displaystyle 1}{\displaystyle r \cdot \sqrt r} \cdot \frac{\displaystyle d \widehat{R}(r)}{\displaystyle dr} +34⋅1r2⋅r⋅R^(r),\displaystyle + \frac{\displaystyle 3}{\displaystyle 4} \cdot \frac{\displaystyle 1}{\displaystyle r^2 \cdot \sqrt r} \cdot \widehat{R}(r),
r2⋅d2R(r)dr2\displaystyle r^2 \cdot \frac{\displaystyle d^2 R(r)}{\displaystyle dr^2} =r⋅r⋅d2R^(r)dr2\displaystyle = r \cdot \sqrt r \cdot \frac{\displaystyle d^2 \widehat{R}(r)}{\displaystyle dr^2} −r⋅dR^(r)dr\displaystyle - \sqrt r \cdot \frac{\displaystyle d \widehat{R}(r)}{\displaystyle dr} +34⋅1r⋅R^(r),\displaystyle + \frac{\displaystyle 3}{\displaystyle 4} \cdot \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot \widehat{R}(r),
r2⋅d2R(r)dr2\displaystyle r^2 \cdot \frac{\displaystyle d^2 R(r)}{\displaystyle dr^2} +2⋅r⋅dR(r)dr\displaystyle + 2 \cdot r \cdot \frac{\displaystyle d R(r)}{\displaystyle dr} =r⋅r⋅d2R^(r)dr2\displaystyle = r \cdot \sqrt r \cdot \frac{\displaystyle d^2 \widehat{R}(r)}{\displaystyle dr^2} +r⋅dR^(r)dr\displaystyle + \sqrt r \cdot \frac{\displaystyle d \widehat{R}(r)}{\displaystyle dr} −14⋅1r⋅R^(r),\displaystyle - \frac{\displaystyle 1}{\displaystyle 4} \cdot \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot \widehat{R}(r),

we divide by r⋅rr \cdot \sqrt r and write equation (G.1) in terms of the coefficients

ζ1⋅d2R^(r)dr2\displaystyle \zeta_1 \cdot \frac{\displaystyle d^2 \widehat{R}(r)}{\displaystyle dr^2} +ζ2⋅dR^(r)dr\displaystyle + \zeta_2 \cdot \frac{\displaystyle d \widehat{R}(r)}{\displaystyle dr} +ζ3⋅R^(r)\displaystyle + \zeta_3 \cdot \widehat{R}(r) =0,\displaystyle = 0,

and now let us determine the coefficients, first transforming the free term

γ2\displaystyle \gamma^2 −k⋅(k+1)r2\displaystyle - \frac{\displaystyle k \cdot (k + 1)}{\displaystyle r^2} −14⋅r2\displaystyle - \frac{\displaystyle 1}{\displaystyle 4 \cdot r^2} =γ2\displaystyle = \gamma^2 −(k+1/2)2r2,\displaystyle - \frac{\displaystyle (k + 1/2)^2}{\displaystyle r^2},
ζ1\displaystyle \zeta_1 =1,ζ2\displaystyle = 1, \quad \zeta_2 =1r,ζ3\displaystyle = \frac{\displaystyle 1}{\displaystyle r}, \quad \zeta_3 =γ2\displaystyle = \gamma^2 −(k+1/2)2r2,\displaystyle - \frac{\displaystyle (k + 1/2)^2}{\displaystyle r^2},

thus, equation (G.1) takes the form

d2R^(r)dr2\displaystyle \frac{\displaystyle d^2 \widehat{R}(r)}{\displaystyle dr^2} +1r⋅dR^(r)dr\displaystyle + \frac{\displaystyle 1}{\displaystyle r} \cdot \frac{\displaystyle d \widehat{R}(r)}{\displaystyle dr} +(γ2−(k+1/2)2r2)⋅R^(r)\displaystyle + \left( \gamma^2 - \frac{\displaystyle (k + 1/2)^2}{\displaystyle r^2} \right) \cdot \widehat{R}(r) =0.\displaystyle = 0.

We have obtained the Bessel equation with half-integer index kk +1/2+ 1/2. Its solutions bounded at zero are the Bessel functions of the first kind R^(r)\widehat{R}(r) =Jk+1/2(γ⋅r)= J_{k + 1/2}(\gamma \cdot r), kk ∈(0..∞)\in (0..\infty); the solutions of the second kind Yk+1/2(γ⋅r)Y_{k + 1/2}(\gamma \cdot r) are infinite at zero, as seen in figure (B.2), and are discarded by the boundedness condition. Thus, the solution of the original equation (G.1) has the form

R(r)\displaystyle R(r) =1r⋅Jk+1/2(γ⋅r),\displaystyle = \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot J_{k + 1/2}(\gamma \cdot r),
(G.2)

where Jk+1/2(γ⋅r)J_{k + 1/2}(\gamma \cdot r) is the Bessel function of the first kind with half-integer index. Up to a constant factor it is the spherical Bessel function: jk(γ⋅r)j_k(\gamma \cdot r) =π2⋅γ⋅r⋅Jk+1/2(γ⋅r)= \sqrt{\frac{\displaystyle \pi}{\displaystyle 2 \cdot \gamma \cdot r}} \cdot J_{k + 1/2}(\gamma \cdot r).