I am a little confused. Let $g(x)$ is a nontrivial solution of the equation $-y'' + q(x)y = \mu y$. Then define (in one book) the operators $$A = g(d/dx) (1/g) \quad and \quad A^* = -(1/g) (d/dx)g$$ (rewrote exactly as in the book)
States that $A^*$ is the formal adjoint of $A$, if $g$ and $\mu$ are real, and that $$A^*A = -\frac{d^2}{dx^2} + q - \mu$$
I don't understand how for example $A$ acts on function: $$Af = g \frac{d}{dx} (f/g) \quad \text{or} \quad Af = g (\frac{d}{dx}\frac{1}{g}) f = g\frac{-g'}{g^2}f \quad?$$
What means "formal"? And how to show that $A^*$ is adjoint for $A$?
- How to show that $$A^*A = -\frac{d^2}{dx^2} + q - \mu$$