To compute the norm of the solution of the equation for the geometry, we need to compute the norm of the spherical Bessel function. It differs from the ordinary Bessel function by the factor 1 r \frac{\displaystyle 1}{\displaystyle \sqrt r} ρ ( r ) \rho(r) = r 2 {} = r^2
where μ \mu d J k + 1 / 2 ( r ) d r ∣ r = μ \frac{\displaystyle d J_{k + 1/2}(r)}{\displaystyle dr} \bigg|_{r=\mu} = 1 2 ⋅ μ ⋅ J k + 1 / 2 ( μ ) {} = \frac{\displaystyle 1}{\displaystyle 2 \cdot \mu} \cdot J_{k + 1/2}(\mu) d d r ( 1 r ⋅ J k + 1 / 2 ( r ) ) ∣ r = μ \frac{\displaystyle d}{\displaystyle dr} \left( \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot J_{k + 1/2}(r) \right) \bigg|_{r=\mu} = 0 {} = 0
Let us try to compute the norm (H.1) ∫ 0 μ ( 1 r ⋅ J k + 1 / 2 ( r ) ) 2 ⋅ r 2 d r \displaystyle \int_0^{\mu} \left( \frac{\displaystyle 1}{\displaystyle \sqrt r} \cdot J_{k + 1/2}(r) \right)^2 \cdot r^2 \,dr = ∫ 0 μ r ⋅ J k + 1 / 2 2 ( r ) d r , \displaystyle {} = \int_0^{\mu} r \cdot J_{k + 1/2}^2(r) \,dr, C.2 ∫ r m + 1 ⋅ J m ( r ) d r \displaystyle \int r^{m+1} \cdot J_m(r) \,dr = r m + 1 ⋅ J m + 1 ( r ) \displaystyle {} = r^{m+1} \cdot J_{m+1}(r) + C , \displaystyle {} + C, C.4 r ⋅ d J m ( r ) d r \displaystyle r \cdot \frac{\displaystyle d J_m(r)}{\displaystyle dr} = m ⋅ J m ( r ) \displaystyle {} = m \cdot J_m(r) − r ⋅ J m + 1 ( r ) , \displaystyle {} - r \cdot J_{m+1}(r), при m \displaystyle \text{при } m = 0 \displaystyle {} = 0 \; ⇒ d J 0 ( r ) d r \displaystyle {} \Rightarrow\; \frac{\displaystyle d J_0(r)}{\displaystyle dr} = − J 1 ( r ) , \displaystyle {} = - J_1(r),
∣ u = r − k − 1 / 2 ⋅ J k + 1 / 2 ( r ) , d v = r k + 3 / 2 ⋅ J k + 1 / 2 ( r ) d r d u = ( − k − 1 2 ) ⋅ r − k − 3 / 2 ⋅ J k + 1 / 2 ( r ) d r d u + r − k − 3 / 2 ⋅ [ ( k + 1 2 ) ⋅ J k + 1 / 2 ( r ) − r ⋅ J k + 3 / 2 ( r ) ] d r d u = − r − k − 1 / 2 ⋅ J k + 3 / 2 ( r ) d r v = r k + 3 / 2 ⋅ J k + 3 / 2 ( r ) ∣ \left | \begin{aligned} &u = r^{-k - 1/2} \cdot J_{k + 1/2}(r), \quad dv = r^{k+3/2} \cdot J_{k + 1/2}(r) \,dr\\ &du = \left( - k - \frac{1}{2} \right) \cdot r^{-k - 3/2} \cdot J_{k + 1/2}(r) \,dr\\ &\phantom{du} + r^{-k - 3/2} \cdot \left[ \left( k + \frac{1}{2} \right) \cdot J_{k + 1/2}(r) - r \cdot J_{k + 3/2}(r) \right] \,dr\\ &du = - r^{-k - 1/2} \cdot J_{k + 3/2}(r) \,dr\\ &v = r^{k+3/2} \cdot J_{k + 3/2}(r) \end{aligned} \right | ∫ 0 μ r ⋅ J k + 1 / 2 2 ( r ) d r \displaystyle \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr = r \displaystyle {} = r ⋅ J k + 1 / 2 ( r ) \displaystyle {} \cdot J_{k+1/2}(r) ⋅ J k + 3 / 2 ( r ) ∣ 0 μ \displaystyle {} \cdot J_{k+3/2}(r) \bigg|_0^{\mu} + ∫ 0 μ r ⋅ J k + 3 / 2 2 ( r ) d r . \displaystyle {} + \int_0^{\mu} r \cdot J_{k+3/2}^2(r) \,dr. Let us transform the eigenvalue equation using formula (C.4 r ⋅ d J m ( r ) d r \displaystyle r \cdot \frac{\displaystyle d J_m(r)}{\displaystyle dr} = m ⋅ J m ( r ) \displaystyle {} = m \cdot J_m(r) − r ⋅ J m + 1 ( r ) , \displaystyle {} - r \cdot J_{m+1}(r), при m \displaystyle \text{при } m = 0 \displaystyle {} = 0 \; ⇒ d J 0 ( r ) d r \displaystyle {} \Rightarrow\; \frac{\displaystyle d J_0(r)}{\displaystyle dr} = − J 1 ( r ) , \displaystyle {} = - J_1(r), r r = μ {} = \mu
μ ⋅ d J k + 1 / 2 ( r ) d r ∣ r = μ \displaystyle \mu \cdot \frac{\displaystyle d J_{k + 1/2}(r)}{\displaystyle dr} \bigg|_{r=\mu} = ( k + 1 2 ) ⋅ J k + 1 / 2 ( μ ) \displaystyle {} = \left( k + \frac{1}{2} \right) \cdot J_{k + 1/2}(\mu) − μ ⋅ J k + 3 / 2 ( μ ) \displaystyle {} - \mu \cdot J_{k + 3/2}(\mu) = 1 2 ⋅ J k + 1 / 2 ( μ ) , \displaystyle {} = \frac{\displaystyle 1}{\displaystyle 2} \cdot J_{k + 1/2}(\mu), k ⋅ J k + 1 / 2 ( μ ) \displaystyle k \cdot J_{k + 1/2}(\mu) = μ ⋅ J k + 3 / 2 ( μ ) , \displaystyle {} = \mu \cdot J_{k + 3/2}(\mu), and substitute the obtained relation into the boundary term
∫ 0 μ r ⋅ J k + 1 / 2 2 ( r ) d r \displaystyle \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr = k ⋅ J k + 1 / 2 2 ( μ ) \displaystyle {} = k \cdot J_{k+1/2}^2(\mu) + ∫ 0 μ r ⋅ J k + 3 / 2 2 ( r ) d r . \displaystyle {} + \int_0^{\mu} r \cdot J_{k+3/2}^2(r) \,dr. We integrate the resulting integral by parts as well:
∣ u = J k + 3 / 2 2 ( r ) , d u = 2 ⋅ J k + 3 / 2 ( r ) ⋅ 1 r ⋅ ( − ( k + 3 / 2 ) ⋅ J k + 3 / 2 ( r ) + r ⋅ J k + 1 / 2 ( r ) ) d r d v = r d r , v = r 2 / 2 ∣ \left | \begin{array}{ll} u = J_{k+3/2}^2(r), &du = 2 \cdot J_{k+3/2}(r) \cdot \frac{\displaystyle 1}{\displaystyle r} \cdot \left( - (k+3/2) \cdot J_{k+3/2}(r) + r \cdot J_{k+1/2}(r) \right) \,dr\\ dv = r \,dr, &v = r^2 / 2 \end{array} \right | ∫ 0 μ r ⋅ J k + 3 / 2 2 ( r ) d r \displaystyle \int_0^{\mu} r \cdot J_{k+3/2}^2(r) \,dr = k 2 2 ⋅ J k + 1 / 2 2 ( μ ) \displaystyle {} = \frac{\displaystyle k^2}{\displaystyle 2} \cdot J_{k+1/2}^2(\mu) + ( k + 3 2 ) \displaystyle {} + \left( k + \frac{3}{2} \right) ⋅ ∫ 0 μ r \displaystyle {} \cdot \int_0^{\mu} r ⋅ J k + 3 / 2 2 ( r ) d r \displaystyle {} \cdot J_{k+3/2}^2(r) \,dr − ∫ 0 μ r 2 \displaystyle {} - \int_0^{\mu} r^2 ⋅ J k + 1 / 2 ( r ) \displaystyle {} \cdot J_{k+1/2}(r) ⋅ J k + 3 / 2 ( r ) d r . \displaystyle {} \cdot J_{k+3/2}(r) \,dr. We integrate the last integral by parts:
∣ u = r k + 5 / 2 ⋅ J k + 1 / 2 ( r ) , d v = r − k − 1 / 2 ⋅ J k + 3 / 2 ( r ) d r d u = ( k + 5 2 ) ⋅ r k + 3 / 2 ⋅ J k + 1 / 2 ( r ) + r k + 3 / 2 ⋅ [ ( k + 1 2 ) ⋅ J k + 1 / 2 ( r ) − r ⋅ J k + 3 / 2 ( r ) ] d r d u = ( 2 ⋅ k + 3 ) ⋅ r k + 3 / 2 ⋅ J k + 1 / 2 ( r ) − r k + 5 / 2 ⋅ J k + 3 / 2 ( r ) d r v = − r − k − 1 / 2 ⋅ J k + 1 / 2 ( r ) ∣ \left | \begin{aligned} &u = r^{k+5/2} \cdot J_{k+1/2}(r), \quad dv = r^{-k-1/2} \cdot J_{k+3/2}(r) \,dr\\ &du = \left( k + \frac{5}{2} \right) \cdot r^{k+3/2} \cdot J_{k+1/2}(r) + r^{k+3/2} \cdot \left[ \left( k + \frac{1}{2} \right) \cdot J_{k+1/2}(r) - r \cdot J_{k+3/2}(r) \right] \,dr\\ &du = (2 \cdot k + 3) \cdot r^{k+3/2} \cdot J_{k+1/2}(r) - r^{k+5/2} \cdot J_{k+3/2}(r) \,dr\\ &v = - r^{-k-1/2} \cdot J_{k+1/2}(r) \end{aligned} \right | ∫ 0 μ r 2 ⋅ J k + 1 / 2 ( r ) ⋅ J k + 3 / 2 ( r ) d r \displaystyle \int_0^{\mu} r^2 \cdot J_{k+1/2}(r) \cdot J_{k+3/2}(r) \,dr = − μ 2 ⋅ J k + 1 / 2 2 ( μ ) \displaystyle {} = - \mu^2 \cdot J_{k+1/2}^2(\mu) + ( 2 ⋅ k + 3 ) \displaystyle {} + (2 \cdot k + 3) ⋅ ∫ 0 μ r \displaystyle {} \cdot \int_0^{\mu} r ⋅ J k + 1 / 2 2 ( r ) d r \displaystyle {} \cdot J_{k+1/2}^2(r) \,dr − ∫ 0 μ r 2 \displaystyle {} - \int_0^{\mu} r^2 ⋅ J k + 1 / 2 ( r ) \displaystyle {} \cdot J_{k+1/2}(r) ⋅ J k + 3 / 2 ( r ) d r , \displaystyle {} \cdot J_{k+3/2}(r) \,dr, ∫ 0 μ r 2 ⋅ J k + 1 / 2 ( r ) ⋅ J k + 3 / 2 ( r ) d r \displaystyle \int_0^{\mu} r^2 \cdot J_{k+1/2}(r) \cdot J_{k+3/2}(r) \,dr = − μ 2 2 ⋅ J k + 1 / 2 2 ( μ ) \displaystyle {} = - \frac{\displaystyle \mu^2}{\displaystyle 2} \cdot J_{k+1/2}^2(\mu) + ( 2 ⋅ k + 3 ) 2 \displaystyle {} + \frac{\displaystyle (2 \cdot k + 3)}{\displaystyle 2} ⋅ ∫ 0 μ r \displaystyle {} \cdot \int_0^{\mu} r ⋅ J k + 1 / 2 2 ( r ) d r . \displaystyle {} \cdot J_{k+1/2}^2(r) \,dr. Substituting the obtained equalities successively into one another, we express the norm:
0 \displaystyle 0 = k 2 2 ⋅ J k + 1 / 2 2 ( μ ) \displaystyle {} = \frac{\displaystyle k^2}{\displaystyle 2} \cdot J_{k+1/2}^2(\mu) + ( k + 1 2 ) \displaystyle {} + \left( k + \frac{1}{2} \right) ⋅ ∫ 0 μ r \displaystyle {} \cdot \int_0^{\mu} r ⋅ J k + 3 / 2 2 ( r ) d r \displaystyle {} \cdot J_{k+3/2}^2(r) \,dr − ∫ 0 μ r 2 \displaystyle {} - \int_0^{\mu} r^2 ⋅ J k + 1 / 2 ( r ) \displaystyle {} \cdot J_{k+1/2}(r) ⋅ J k + 3 / 2 ( r ) d r , \displaystyle {} \cdot J_{k+3/2}(r) \,dr, ∫ 0 μ r ⋅ J k + 3 / 2 2 ( r ) d r \displaystyle \int_0^{\mu} r \cdot J_{k+3/2}^2(r) \,dr = − k 2 2 ⋅ k + 1 ⋅ J k + 1 / 2 2 ( μ ) \displaystyle {} = - \frac{\displaystyle k^2}{\displaystyle 2 \cdot k + 1} \cdot J_{k+1/2}^2(\mu) + 2 2 ⋅ k + 1 \displaystyle {} + \frac{\displaystyle 2}{\displaystyle 2 \cdot k + 1} ⋅ ∫ 0 μ r 2 \displaystyle {} \cdot \int_0^{\mu} r^2 ⋅ J k + 1 / 2 ( r ) \displaystyle {} \cdot J_{k+1/2}(r) ⋅ J k + 3 / 2 ( r ) d r , \displaystyle {} \cdot J_{k+3/2}(r) \,dr, ∫ 0 μ r ⋅ J k + 1 / 2 2 ( r ) d r \displaystyle \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr = k ⋅ ( k + 1 ) 2 ⋅ k + 1 \displaystyle {} = \frac{\displaystyle k \cdot (k+1)}{\displaystyle 2 \cdot k + 1} ⋅ J k + 1 / 2 2 ( μ ) \displaystyle {} \cdot J_{k+1/2}^2(\mu) + 2 2 ⋅ k + 1 \displaystyle {} + \frac{\displaystyle 2}{\displaystyle 2 \cdot k + 1} ⋅ ∫ 0 μ r 2 \displaystyle {} \cdot \int_0^{\mu} r^2 ⋅ J k + 1 / 2 ( r ) \displaystyle {} \cdot J_{k+1/2}(r) ⋅ J k + 3 / 2 ( r ) d r , \displaystyle {} \cdot J_{k+3/2}(r) \,dr, ∫ 0 μ r ⋅ J k + 1 / 2 2 ( r ) d r \displaystyle \int_0^{\mu} r \cdot J_{k+1/2}^2(r) \,dr = k ⋅ ( k + 1 ) − μ 2 2 ⋅ k + 1 \displaystyle {} = \frac{\displaystyle k \cdot (k+1) - \mu^2}{\displaystyle 2 \cdot k + 1} ⋅ J k + 1 / 2 2 ( μ ) \displaystyle {} \cdot J_{k+1/2}^2(\mu) + 2 ⋅ k + 3 2 ⋅ k + 1 \displaystyle {} + \frac{\displaystyle 2 \cdot k + 3}{\displaystyle 2 \cdot k + 1} ⋅ ∫ 0 μ r \displaystyle {} \cdot \int_0^{\mu} r ⋅ J k + 1 / 2 2 ( r ) d r , \displaystyle {} \cdot J_{k+1/2}^2(r) \,dr, k ⋅ ( k + 1 ) − μ 2 2 ⋅ k + 1 ⋅ J k + 1 / 2 2 ( μ ) \displaystyle \frac{\displaystyle k \cdot (k+1) - \mu^2}{\displaystyle 2 \cdot k + 1} \cdot J_{k+1/2}^2(\mu) = − 2 2 ⋅ k + 1 \displaystyle {} = - \frac{\displaystyle 2}{\displaystyle 2 \cdot k + 1} ⋅ ∫ 0 μ r \displaystyle {} \cdot \int_0^{\mu} r ⋅ J k + 1 / 2 2 ( r ) d r . \displaystyle {} \cdot J_{k+1/2}^2(r) \,dr. Finally we obtain:
G. The Bessel equation in spherical coordinates I. The Bubnov–Galerkin functional