Eigenfunctions of \hat{O}: Find Solutions Here

[0xDEADBEEF]

Hi there,

I don't have Mathematica and the standard books didn't work. I am looking for the eigenfunctions to this operator:

$$\hat{O} = (\sin \alpha) \, \partial_x +(\cos \alpha) \, x \qquad \alpha \in \mathbb{R}$$

 

[Mute]

Well, it's a first order equation, so you can solve it using an integrating factor.

$$[(\sin \alpha) \, \partial_x +(\cos \alpha) \, x]\psi(x) = \lambda \psi(x)$$

and so moving the RHS over to the LHS and multiplying by $\mu(x)$, and dividing the $\sin \alpha$ through,

$$\mu(x)\partial_x \psi(x) + \left[(\cot \alpha) \, x - \frac{\lambda}{\sin\alpha}\right] \psi(x)\mu(x) = 0$$

Hence through the usual arguments let

$$\frac{d\mu}{dx} = \left[(\cot \alpha) \, x - \frac{\lambda}{\sin\alpha}\right] \mu$$

and the above expression is just the product rule expanded and you can solve for $\psi$. Then apply the initial condition, and possibly any other conditions (such as demanded the function not diverge as x gets large) that might put restrictions on the values of $\lambda$.

 

[0xDEADBEEF]

Thanks a lot!