Math Is Fun Forum

Related to a physical problem in Quantum Mechanics the following differential equation arises:

(l (l + 1) + b (x^3) ((x^(3/4))/((-1 + x)^(7/4))) - 2 b (x^2)) y - 2 x y' + x (-1 + x) y'' == 0

with the border conditions:

Limit[y[x], x -> 1] = 0

and

Limit[y[x], x -> Infinity] = 0

and we can set y[x]=0 for all x<1 (that means, for x<1 a solution other than the trivial solution y[x]=0 makes no physical sense)

The term l(l+1) corresponds to an angular momentum with l=...-3,-2,-1,0,1,2,3...

And b is an energy related Eigenvalue, perhaps depending on l and additional integer numbers (quantum numbers).

This differential equation has an irregular singularity at x=1 . That means, there is no standard method, for example like the Frobenius method, to solve this equation, because the solution will provide terms, which can not be developed in a Taylor series at x=1. A term with this property would be for example E^(-1/(x-1)).

The problem of this differential equation for me is, that I could not find any transformation to a known and solved differential equation. The numerical solution shows a shape of the function as expected. The ground state will provide a function starting from y[1]=0 (boundary condition) rising to a maximum and will finally fall down to 0 (limit to Infinity). I can imagine the excited states. The first excited state will show a similar behaviour at x=1 and infinity, but the function will cross the x-axis. The second excited state will cross the x-axis two times and so on.

This differential equation describes matter in a new kind of understanding. I am convinced, that this problem will be solved by intuition. Mathematica and Wolfram Alpha failed. Solving this differential equation means, to find the right trick. For this I give it a try in this forum. Perhaps there is someone who is able "to see" the right way.

Regards

Robert