Help with Eigenvalue Equation and Fourier Transform

[Helloaksdoq]

Homework Statement

Homework Equations

The Attempt at a Solution

I did Fourier transform directly to the eigenvalue equation and got

Psi(p)=a*Psi(0)/(p^2/2m-E)

But the rest, I don't even know where to start.
Any opinion guys?

 

[Goddar]

Hi. What you got so far seems right.
Now i can't think of any easy "trick" to show the uniqueness of E in this representation, so i would suggest you work on how to enforce:
<φ_E(p)|φ_E'(p)> = δ(E–E')
I suppose you can look up the integral; given the result you should see:
1 - how E needs to be positive
2 - how E must be = E' (i.e.: E is unique)
That's a start...