I am dealing with the Schrodinger operator, $$S = -\frac{d^2}{dx^2}+p(x).$$
Let $f$ be a function satisfying $$Sf = k^2 f, \ \ \ k^2\in\mathbb{R} \ \ \ \ (1)$$ also let $g$ satisfy this same equation.
I have the expression $\int(fSg)dx$ but I would like to 'swap' $f$ and $g$ in the integral. That is I want to express it as $S$ acting on something to do with $f$ rather than $g$. Here is my attempt:
Let $(f,g) = \int{\bar{f} g}dx$ be the inner product.
Then $\int(fSg)dx = (\bar{f},Sg)$ but now using that $S$ is symmetric, $$(\bar{f}, Sg) = \overline{(g, S\bar{f})} = \overline{\int \bar{g} S\bar{f}dx} $$
Note that $S\bar{f} = k^2 \bar{f}$ since $f$ satisfies $(1)$ above. Putting it all together then we get $$(\bar{f}, Sg) = k^2\int gf dx$$
Are these manipulations correct?
\begin{eqnarray*} \int (fSg) dx= \int (f(x) k^2 g(x)) dx= \int (g(x) k^2 f(x)) dx = \int (gSf) dx \end{eqnarray*}