Asymptotic form of a wave function

[iamalexalright]

Homework Statement

For one of my physics assignments I need to do some numeric integration (and I know its physics but my question doesn't pertain to that).

Given
$\hat{H}\Phi = E\Phi$
$\frac{\delta^{2}\Phi}{\delta x^{2}}(x) = \frac{-2m}{h^{2}}(E - V(x))\Phi(x) = -Z(x)\Phi(x)$

We have the taylor expansion:
$\Phi(x + 2d) = \Phi(x) + 2d\frac{\delta \Phi}{\delta x} + 2d^{2}\frac{\delta^{2}\Phi}{\delta x^{2}}$

Using definition of the derivative:
$\frac{\delta \Phi(x)}{\delta x} = lim_{d\rightarrow 0} \frac{\Phi(x + d) - \Phi(x - d)}{2d}$

we can obtain:
$\Phi(x + 2d) = \Phi(x + d) + \Phi(x) - \Phi(x - d) - 2d^{2}Z(x)\Phi(x)$.

For the potential V(x) = Ax with A = -1, m = 1, h = 1, and E = 0, find the asymptotic form of the wave function for x >> 0. Use this form to find the three initial points.


This is where I run into problems. Maybe I'm missing something very obvious but I don't know how to find this initial asymptotic form. Any hints?

 

[lippyka]

Homework Equations None really, just the ones listed above.The Attempt at a Solution I've tried rewriting the equations in different ways but I don't know where to go from here.