Solving eigenvectors of Operator (a+)^2-(a)^2

[AnnaKodanev]

Homework Statement

I need to find the eigenvectors of the following operator (a+)^2-(a)^2, when (a+), (a) are the creation and the annihilation operators.

Homework Equations

The Attempt at a Solution

I tried to put the eigenvectors as sum of eigenvectors of operator N=(a+)(a).Maybe you know some tricks that can simplify the solution?

 

[Tomsk]

I think you need to write (a+) and (a) as -d/dy + y and d/dy + y, where $x=\alpha y$, and $$\alpha=\left(\frac{\hbar}{\sqrt{mk}}\right)^{1/2}$$. Then write out (a+)^2-(a)^2, and solve as a differential equation.

 

[AnnaKodanev]

Thank you. I will try.