Lets say I have a variable $x\in [0,1]$ and I start from a differential Operator $O_1=f(x)D^2+g(x)$ where $f$ and $g$ are well behaved functions on $x\in[0,1]$. Also consider another well behaved function $h(x)$ on the same interval.
Now, suppose I want to determine the follwing in a functional sense:
$$Tr[O_1^{-1}h]=Tr\left[\frac{h(x)}{f(x)D^2+g(x)}\right]=Tr\left[\frac{1}{f(x)}D^{-2}\frac{h(x)}{1+D^{-2}\frac{g(x)}{f(x)}}\right]=?$$
Where now $D^{-2}$ is an antiderivative (integration) operator. Currently I have no idea how to proceed. Any suggestions and references are welcome!