What's the standard name of this equation so i can look up how to solve it?

[bjnartowt]

what's the standard "name" of this equation so i can look up how to solve it?

Homework Statement

Find the solution to

$$\left( {\frac{1}{{{r^2}}}\frac{d}{{dr}}\left( {{r^2}\frac{d}{{dr}}} \right) - \frac{{\ell (\ell + 1)}}{{{r^2}}} + {k^2}} \right){g_{k\ell }}(r,r') = - \frac{1}{{{r^2}}}\delta (r - r')$$

subject to the boundary conditions: $${g_{k\ell }}(0,r') = 0{\rm{ and }}{g_{k\ell }}(r,r')\~{\textstyle{1 \over r}}\exp ({\bf{i}}kr){\rm{ for large r}}$$

Homework Equations

see problem statement.

The Attempt at a Solution

I think this is "of the form",
$$L{g_{k\ell }}(r,r') = \delta (r - r')$$

...where L is a linear operator. What is the "name" of this equation (e.g., is it an "inhomogeneous linear ODE"?). I need to know the "name" so I can look up the solution method somewhere. (I didn't have the best differential equations course).

***see attached .pdf for LaTeX stuff that got garbled...***

 

[hunt_mat]

Bessel's function in spherical form?

 

[vela]

The spherical Bessel functions are the solutions to the homogeneous differential equation with that linear operator. The righthand side is the Dirac delta function in spherical coordinates with the angular dependence integrated out.

As far as equations of the form Lg(r,r')=-δ(r-r'), it looks like that's what you're pretty much learning to solve now. You want to look into the topic of Green's functions.

 

[bjnartowt]

Hi, thank you for your responses.

Many people are telling me that the solution to this equation is a spherical Bessel function, but I cannot yet believe that. Please see attached .pdf...

 

[vela]

You might find the post by HallsofIvy in this thread enlightening. He illustrates the general method of finding a Green's function, albeit for a much simpler example.

You'll need to find the proper amount of the jump in the derivative for your problem.