- #1
art1915
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- TL;DR
- Solving 2nd order ODE with rank 3 irregular singularity
I am searching for a solution ##R(r)## of the following radial par1 of the Klein-Gordon equation $$[1]\qquad -\left(1-\left(\frac{r_0}{r}\right)^3\right)^{\frac{4}{3}} \partial_r^2 R(r) - \frac{2}{r \left(1-\left(\frac{r_0}{r}\right)^3\right)^{\frac{2}{3}}} \partial_r R(r) + \left(\frac{l(l+1)}{r^2 \left(1-\left(\frac{r_0}{r}\right)^3\right)^{\frac{2}{3}}} + \mu \right) R(r) = 0$$
* with ##R:\mathbb{R} \to \mathbb{C},r \to R(r)##
* with the constant ##r_0 \in \mathbb{R} \wedge r_0 \gt 0## which is a radius with the dimension ##m## (meter)
* the solution extends across two regions, an "inner range" from ##0 \le r \le r_0 ## and an "outer range" ##r_0 \le r##
* and with the constant ##\mu \in \mathbb{R}## with dimension ##m^{-2}##
* and with the dimensionless constant ##l=0,1,2,3...##
* and with the boundary condition ##\lim \limits_{r \to r_0} R(r) = 0## for the „inner range“, which can be called the boundary condition for the „inner solution“
* and with the boundary condition ##\lim \limits_{r \to \infty} R(r) = 0## for the „outer range“, which can be called the boundary condition for the „outer solution“
With the transformation ##u(r)=(r^3 - r_0^3)## equation [1] will be transformed to $$[2]\qquad \partial_u^2 R(u) + \frac{2}{u}\left(1 + \frac{r_0^6}{u^3 \left(u^3 + r_0^3\right)}\right) \partial_u R(u) - \left(\frac{l(l+1)}{u^2} + \mu \right) R(u) = 0$$
Equation [2] can be rearranged to $$[2a]\qquad \partial_u^2 R(u) + 2\left(\frac{r_0^3}{u^4} + \frac{u^2}{u^3 + r_0^3}\right) \partial_u R(u) - \left(\frac{l(l+1)}{u^2} + \mu \right) R(u) = 0$$
For ##r_0=0## equation [2] will turn to the Spherical Bessel Equation
https://www.damtp.cam.ac.uk/user/tong/aqm/bessel.pdf
which plays an important role in physics, even for the "particle in a sphere" (see page 16)
https://www.diva-portal.org/smash/get/diva2:815669/FULLTEXT01.pdf
Because of the roots of ##u^3+r_0^3=\left(u+r_0\right)\left(u+r_0\frac{1}{2}\left(1+i\sqrt{3}\right)\right)\left(u+r_0\frac{1}{2}\left(1-i\sqrt{3}\right)\right)## equation [2] has regular singularities at ##u_1=-r_0##, ##u_2=-r_0\frac{1}{2}\left(1+i\sqrt{3}\right)## and ##u_3=-r_0\frac{1}{2}\left(1-i\sqrt{3}\right)## and an irregular singularity of ##rank\ 3## at ##u_0=0## and an irregular singularity of ##rank\ 3## at ##u_4=\infty## (see Möbius transformation with ##u=w^{-1}##).
The inner solution
The irregular singularity at ##u_0=0## indicates, that a Frobenius solution is not possible. Instead one has to use a different Ansatz, as mentioned on page 32 in the mathematics lecture "Ordinary Differential Equations" from Henry Liu at the Columbia University:
https://member.ipmu.jp/henry.liu/teaching/ss19-ode/notes.pdf
For this, the Ansatz for a solution ##y(x)## covering an irregular singularity ##x_0## of ##rank\ k## should be $$y(x)=y_0(x)\ exp\left(\frac{c_{k}}{(x-x_0)^{k}} + \frac{c_{k-1}}{(x-x_0)^{k-1}} + ... + \frac{c_{1}}{x-x_0}\right)$$ where ##y_0(x)## is a Frobenius solution.
The inner solution of [2] should look like $$R(u)=R_0(u)\ exp\left(\frac{c_{3}}{u^3} + \frac{c_{2}}{u^2} + \frac{c_{1}}{u}\right)$$ where ##R_0(u)## is a Frobenius solution.
Questions
(1) How should I prepare the Ansatz for the Frobenius solution ##R_0(u)## with respect to the regular singularities at ##u_1=-r_0##, ##u_2=-r_0\frac{1}{2}\left(1+i\sqrt{3}\right)## and ##u_3=-r_0\frac{1}{2}\left(1-i\sqrt{3}\right)## for the inner solution?
(2) Is there any other option to solve equation [2], perhaps with a transformation into another kind of 2nd order ODE which is perhaps easier to solve?
(3) is there a method to determine the dependency or restrictions on "quantum numbers" of ##\mu## and/or ##r_0## without an explicit solution ##R(u)##?
* with ##R:\mathbb{R} \to \mathbb{C},r \to R(r)##
* with the constant ##r_0 \in \mathbb{R} \wedge r_0 \gt 0## which is a radius with the dimension ##m## (meter)
* the solution extends across two regions, an "inner range" from ##0 \le r \le r_0 ## and an "outer range" ##r_0 \le r##
* and with the constant ##\mu \in \mathbb{R}## with dimension ##m^{-2}##
* and with the dimensionless constant ##l=0,1,2,3...##
* and with the boundary condition ##\lim \limits_{r \to r_0} R(r) = 0## for the „inner range“, which can be called the boundary condition for the „inner solution“
* and with the boundary condition ##\lim \limits_{r \to \infty} R(r) = 0## for the „outer range“, which can be called the boundary condition for the „outer solution“
With the transformation ##u(r)=(r^3 - r_0^3)## equation [1] will be transformed to $$[2]\qquad \partial_u^2 R(u) + \frac{2}{u}\left(1 + \frac{r_0^6}{u^3 \left(u^3 + r_0^3\right)}\right) \partial_u R(u) - \left(\frac{l(l+1)}{u^2} + \mu \right) R(u) = 0$$
Equation [2] can be rearranged to $$[2a]\qquad \partial_u^2 R(u) + 2\left(\frac{r_0^3}{u^4} + \frac{u^2}{u^3 + r_0^3}\right) \partial_u R(u) - \left(\frac{l(l+1)}{u^2} + \mu \right) R(u) = 0$$
For ##r_0=0## equation [2] will turn to the Spherical Bessel Equation
https://www.damtp.cam.ac.uk/user/tong/aqm/bessel.pdf
which plays an important role in physics, even for the "particle in a sphere" (see page 16)
https://www.diva-portal.org/smash/get/diva2:815669/FULLTEXT01.pdf
Because of the roots of ##u^3+r_0^3=\left(u+r_0\right)\left(u+r_0\frac{1}{2}\left(1+i\sqrt{3}\right)\right)\left(u+r_0\frac{1}{2}\left(1-i\sqrt{3}\right)\right)## equation [2] has regular singularities at ##u_1=-r_0##, ##u_2=-r_0\frac{1}{2}\left(1+i\sqrt{3}\right)## and ##u_3=-r_0\frac{1}{2}\left(1-i\sqrt{3}\right)## and an irregular singularity of ##rank\ 3## at ##u_0=0## and an irregular singularity of ##rank\ 3## at ##u_4=\infty## (see Möbius transformation with ##u=w^{-1}##).
The inner solution
The irregular singularity at ##u_0=0## indicates, that a Frobenius solution is not possible. Instead one has to use a different Ansatz, as mentioned on page 32 in the mathematics lecture "Ordinary Differential Equations" from Henry Liu at the Columbia University:
https://member.ipmu.jp/henry.liu/teaching/ss19-ode/notes.pdf
For this, the Ansatz for a solution ##y(x)## covering an irregular singularity ##x_0## of ##rank\ k## should be $$y(x)=y_0(x)\ exp\left(\frac{c_{k}}{(x-x_0)^{k}} + \frac{c_{k-1}}{(x-x_0)^{k-1}} + ... + \frac{c_{1}}{x-x_0}\right)$$ where ##y_0(x)## is a Frobenius solution.
The inner solution of [2] should look like $$R(u)=R_0(u)\ exp\left(\frac{c_{3}}{u^3} + \frac{c_{2}}{u^2} + \frac{c_{1}}{u}\right)$$ where ##R_0(u)## is a Frobenius solution.
Questions
(1) How should I prepare the Ansatz for the Frobenius solution ##R_0(u)## with respect to the regular singularities at ##u_1=-r_0##, ##u_2=-r_0\frac{1}{2}\left(1+i\sqrt{3}\right)## and ##u_3=-r_0\frac{1}{2}\left(1-i\sqrt{3}\right)## for the inner solution?
(2) Is there any other option to solve equation [2], perhaps with a transformation into another kind of 2nd order ODE which is perhaps easier to solve?
(3) is there a method to determine the dependency or restrictions on "quantum numbers" of ##\mu## and/or ##r_0## without an explicit solution ##R(u)##?