It's very much interesting but don't know about it.

@P.Contreras

I saw your reply .Potential is continuous and bottomless in nature .vIt contains no singularity except at |x| = infinity.

Why the wave function, which is in the form of bessel function be zero at x=0 ,so also its derivative.

In general wave function is zero at infinity ,as potential is infinity .

I feel wave function zero at x=0 , is an adhoc condition. Why can't it be a constant say c.

Thank you for sharing your valuable reference .

@P.Contreras

Thank you for your references.

Soon I will be touch with you.

@ P.Contreras

A little idea comes to mind on

1. V(x)= - V ( exp(2|x|/a) -1)

2.V(x)=- V(exp(2x/a) -1) for x>0 only

V(x)=0 for x