Given the Hamiltonian $\hat{H}=\hat{x}\hat{p}^2\hat{x}$ with position operator $\hat{x}$ and momentum operator $\hat{p}$, I should give an explicit expression for the measure of the path integral, $$ \int\mathcal{D}[x(t)]\exp\left\{\frac{i}{\hbar}S[x(t)]\right\}, $$ wherein $$ S[x(t)]=\int dt\mathcal{L}(x,\dot{x}), $$ is the action with $\mathcal{L}$ being the Lagrangian.
I thought of two approaches on how to find an explicit expression for the path integral:
1. Reverse the Legendre transformation to find the Lagrangian
We first need to relate the momentum operator with the time derivative of the position operator, e.g. $$ \frac{d\hat{x}}{dt}=[\hat{x},\hat{H}]=2i\hbar\hat{p}, $$ using the canonical commutation relation $[\hat{x},\hat{p}]=i\hbar$.
Finally, the Lagrangian is obtained via, $$ \mathcal{L}(\hat{x},d\hat{x}/dt)=\hat{p}\frac{d\hat{x}}{dt}-H(\hat{x},d\hat{x}/dt)=2i\hbar\left(\frac{1}{2i\hbar}\frac{d\hat{x}}{dt}\right)^2-\hat{x}\left(\frac{1}{2i\hbar}\frac{d\hat{x}}{dt}\right)\hat{x}. $$
I don't know how to proceed from here. Inserting the Lagrangian into the action won't give us any new insights.
2. Derive the path integral formalism from the time evolution operator
This approach is inspired by the derivation used by Shankar in "Principles of Quantum Mechanics 2nd Edition, p. 585".
We start with the time evolution operator in position represenation, $$ U(x_0,x_n,t)=\langle x_0\mid \exp\left[\frac{it}{\hbar}\hat{H}\right]\mid x_n\rangle, $$ and define $\epsilon=t/N$ where we assume $\epsilon\ll1$, thus, $$ U(x_0,x_n,t)=\langle x_0\mid \exp\left[\frac{i\epsilon}{\hbar}\hat{H}\right]^N\mid x_n\rangle=\langle x_0\mid \exp\left[\frac{i\epsilon}{\hbar}\hat{H}\right]\dots\exp\left[\frac{i\epsilon}{\hbar}\hat{H}\right]\mid x_n\rangle+\mathcal{O}(\epsilon^2), $$ where we have used the Baker-Campbell-Hausdorff formula and neglected higher orders of $\epsilon$.
We can now use the completeness identity for the position operator to obtain, $$ U(x_0,x_n,t)=\int dx_1\dots dx_{n-1}\langle x_0\mid \exp\left[\frac{i\epsilon}{\hbar}\hat{H}\right]\mid x_1\rangle\dots\langle x_{n-1}\mid\exp\left[\frac{i\epsilon}{\hbar}\hat{H}\right]\mid x_n\rangle+\mathcal{O}(\epsilon^2). $$
Using the completeness relation in momentum space, we can give an expression for the matrix elements, $$ \langle x_{n-1}\mid\exp\left[\frac{i\epsilon}{\hbar}\hat{H}\right]\mid x_n\rangle=\sum^\infty_{m=0}\frac{(i\epsilon/\hbar)^m}{m!}\langle x_{n-1}\mid\hat{x}\hat{p}^2\hat{x}\mid x_n\rangle=\sum^\infty_{m=0}\frac{(i\epsilon/\hbar)^m}{m!}x_nx_{n-1}\delta^{(2m)}(x_n-x_{n-1}), $$ wherein $\delta^{(2m)}$ denotes the $2m$ derivative with respect to the argument of the $\delta$ distribution.
However, this expression does not appear to be of much help.
Are these approaches the any good? How do I find an explicit expression for the measure of the path integral?
