Computational solution of exponential decaying wavefunction tail

As I am going through some (quite simple) computational physics exercices I have a question concerning one exercise that involves solving the radial Schrodinger equation.

This is done with the Numerov method (discrete integrator). This gives fairly good results, for every example I try where the energy of my wave is higher than the used potential.

But when my wavefunction enters an area where (V-E) > 0, we should expect a decaying exponential tail. My computer program gives this (to some extent) but after a few (rescaled) length units I get an exponential growth that blows up my solution.

I know the solution to ( symbol " denotes double derivative) : X" = +k^2 X gives as basic solutions

A exp(kx) + B exp(-kx)

but the exponentially growing solutions are non physical and only the decaying solution is kept. Is it common for computational solutions to this equation to mix the exponentially growing solution into this? Is there a way to avoid this from happening? Is this due to the used algorithm (Numerov)? Is this also common in other used methods to solve this problem?

Or do you guys think my program is just plain wrong?

(It gives all the expected results for numerous tried examples, and it is now almost copied from the standard solution given by my professor at my university so I shouldn't see why it is wrong).

• well I just answered this physics.stackexchange.com/a/452453/36194 which might have some elements of solution for you. – ZeroTheHero Jan 6 at 20:52
• That entirely answers my question. Answer on point. Thank you very much! I was also trying the harmonic oscillator and had the same results. Only thing I can maybe think of to help this problem is integrate backwards from very large distance r? – CFRedDemon Jan 6 at 21:19
• if you integrate from $\infty$ you’ll get in trouble near $0$...I don’t know any escape beyond finer mesh and higher order schemes, both of which imply longer run times. – ZeroTheHero Jan 6 at 21:27