Schrodinger equation in momentum space- How to make it? 
Use the Schrodinger equation in momentum space to analyze bound states of a particle in the potential, $U(x)=α[δ(x-a)+δ(x+a)]$.

I'm a student in a university and I'm learning about Schrödinger's equation. I wonder how to get it in momentum space. The hint of this problem show that:
 
I would like to ask that how to get a result for U(p) and take the form of the schrodinger in momentum space like the hint. I really need your help!!
 A: I'll give a hand which should elucidate how to solve the problem. Recall the Schrodinger equation is
$$\left(-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)\right)\psi(x) = E\psi(x).$$
The Fourier transformed wave function $\tilde{\psi}(p)$ is given by,
$$\tilde{\psi}(p)=\int dx \, e^{-ipx}\psi(x).$$
In finding the momentum space version of the equation, we have to compute the integral,
$$\mathcal{F}\{V(x)\psi(x)\} = \alpha \int dx \, e^{-ipx}\left[ \delta(x-a) + \delta(x+a)\right]\psi(x)=\alpha(\psi(a)e^{-ipa}+\psi(-a)e^{ipa})$$
using the rules of integrating delta functions. We thus arrive at,
$$\frac{\hbar^2p^2}{2m}\tilde{\psi}(p) + \alpha(\psi(a)e^{-ipa}+\psi(-a)e^{ipa}) = E\tilde{\psi}(p).$$
A: You can take the equation, and do a Fourier Transform on that. 
Say (for simplicity) in 1-D, we have,
$$-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\psi(x)=(E-V(x))\psi(x)$$
Now you can express the wave function in terms of its Fourier Transform,
$$\psi(x)=\frac{1}{2\pi}\int dk e^{-ikx}\,\tilde{\psi}(k)$$
and similarly for the potential $V(x)$. Plug these into the Schrödinger equation, and you can work out the equation in momentum representation.
