I'm very familiar with solving differential equations. I think I'm just struggling with the setup here because I've never done it with operators. I've always just been given the differential equation and told to find the eigenfunctions. In this physics class, I have to setup the differential equation.
Note: $p = -i \frac{d}{dx}$, the momentum operator.
The operator I'm given is $H=x^3p + px^3$
I am told to show it is formally self-adjoint then to find the eigenfunction and show it has purely imaginary eigenvalues.
(The problem says the eigenfunction is $|x|^{-3/2}\exp(\lambda/4x^2)$, but I have to show that is the case, and I'd like to know how to find it myself.)
I think I am likely setting it up wrong. From what I've seen online, you set the ODE equal to $\lambda * x$.
$-x^3 i \frac{d}{dx}
– i \frac{d}{dx} x^3 = \lambda x$
$-x^3 i \frac{d}{dx} – i3x^2 = \lambda x$
$\frac{d}{dx} + \frac{3}{x} = – \frac{\lambda}{ix^2}$
I figure from here I could separate it, so I wrote d/dx as dy/dx.
$\frac{dy}{dx} = -\frac{3}{x}- \frac{\lambda}{ix^2}$
$dy = \left(-\frac{3}{x} – \frac{\lambda}{ix^2}\right) dx$
$y = -3\ln|x| + \frac{\lambda}{ix}$
If you e both sides, I got very close
$|x|^{-3}e^{\lambda/xi}$, but this is not exactly right. As I've said many times, I'm pretty sure it is my setup, can anyone guide me on how to set this problem up?
Best Answer
You must apply the operator to a function.
$$\left(ix^3\frac d{dx}-i\frac d{dx}x^3\right)y=-2ix^3\frac{dy}{dx}-3ix^2y=\lambda y$$
is a separable equation.
$$\frac{y'}y=i\frac\lambda{4x^3}-\frac3{2x}$$
and
$$y=c x^{3/2}\exp\left(-i\frac\lambda{4x^2}\right)$$
Credits to @Gregory.