Successful reformulation of Path Integral Method?

Question

Question
The following is an attempt to reformulate Feynman's path integral formulation. Is this correct? Can this be made rigorous?  What is the radius of convergence? 

Proof
Let's think about Feynman's Path Integral Formulation of Quantum Mechanics.

$$A = \langle x_i|e^{\frac{i}{\hbar} \int H dt}| x_f\rangle$$

Splitting into infinitesimally many $\delta t$:

$$A = \langle x_i|\prod_j e^{\frac{i}{\hbar} H \delta t_j}| x_f\rangle$$

Inserting infinitely many ($N -1 \nearrow \infty$) identity operators of the form $\int | x_{t_k}\rangle\langle x_{t_k}|  x_{t_k}$

$$A = \langle x_i|e^{\frac{i}{\hbar} H \delta t_j} \Big(\prod_{j=1}^{N-1} \int |x_{t_j}\rangle \langle x_{t_j}| d x_{t_j}e^{\frac{i}{\hbar} H \delta t_j}\Big)| x_f\rangle$$

Now, we stop at this step and pull out a random $x_{t_k}$:

$$A =  \int \int \langle x_i|e^{\frac{i}{\hbar} H \delta t_j}  \Big(\prod_{j\neq k} \int|x_{t_j}\rangle \langle x_{t_j}| d x_{t_j}e^{\frac{i}{\hbar} H \delta t_j}\Big)| x_f\rangle \int \langle x_{t_{k-1}}| e^{i H \delta t_k} | x_{t_{k}} \rangle \langle  x_{t_{k}}| e^{i H \delta t_k} | x_{t_{k+1}} \rangle  dx_{t_{k}} d x_{t_{k-1}} d x_{t_{k+1}}$$

Adding over all possible $t_k$'s

$$N A = \sum_{k} \int \int  \langle x_i|e^{\frac{i}{\hbar} H \delta t_j}  \Big(\prod_{j\neq k} \int|x_{t_j}\rangle \langle x_{t_j}| d x_{t_j}e^{\frac{i}{\hbar} H \delta t_j}\Big)| x_f\rangle \int \langle x_{t_{k-1}}| e^{i H \delta t_k} | x_{t_{k}} \rangle \langle  x_{t_{k}}| e^{i H \delta t_k} | x_{t_{k+1}} \rangle  dx_{t_{k}} d x_{t_{k-1}} d x_{t_{k+1}}$$

Since, $k$ is just a dummy index we remove them (with $\delta t_l = \delta t_m$ where $l$ and $m$ are arbitrary)  and proceed to use $N \delta t= t_f -t_i =T$ (where $t_f$ is the final time and $t_i$ is the initial time):

$$T A = \sum_{k}  \int \int \Big(  \langle x_i|e^{\frac{i}{\hbar} H \delta t_j}  \Big(\prod_{j\neq k} \int|x_{t_j}\rangle \langle x_{t_j}| d x_{t_j}e^{\frac{i}{\hbar} H \delta t_j}\Big)| x_f\rangle \int \langle x_{t_{k-1}}| e^{i H \delta t_k} | x_{t_{k}} \rangle \langle  x_{t_{k}}| e^{i H \delta t_k} | x_{t_{k+1}} \rangle  dx_{t_{k}} \Big ) \delta t_k  d x_{t_{k-1}} d x_{t_{k+1}}$$

We make the redefinition:

$$\Big(\prod_{j\neq k} \int|x_{t_j}\rangle \langle x_{t_j}| d x_{t_j}e^{\frac{i}{\hbar} H \delta t_j}\Big) = a_k$$

Now, swapping the summation and the $2$ integrals

$$T A = \int \int \sum_{k=1}^{N-1} a_k \int \langle x_{t_{k-1}}| e^{i H \delta t_k} | x_{t_{k}} \rangle \langle  x_{t_{k}}| e^{i H \delta t_k} | x_{t_{k+1}} \rangle  dx_{t_{k}}  \delta t_k  d x_{t_{k-1}} d x_{t_{k+1}}$$

Evaluating the $\delta t_k$ integral first using this:

$$T A = \int \int \sum_{k=1}^{\infty} \lim_{s \to 1} \frac{a_k}{k^s} \times \frac{1}{\zeta(s)} \int \int_{t_i}^{t_f} \langle x_{t_{k-1}}| e^{i H \delta t_k} | x_{t_{k}} \rangle \langle  x_{t_{k}}| e^{i H \delta t_k} | x_{t_{k+1}} \rangle  \delta t_k   dx_{t_{k}}   d x_{t_{k-1}} d x_{t_{k+1}}$$

Comments

Partial answer: It deppends a lot on what is $H$. First try with harmonic oscilators... and do all the path...

Something that can be very useful for you are the next references:

If you are interested in rigorous formulations of path integral method, there are a lot, but Sergio Albeverio work on path integrals for QM is great, his book with Raphael Høegh-Krohn "Mathematical Theory of Feynman Path Integrals" is very good.

Also, path integral for QM has been done rigorous in a lot of instances, what's more messy is path integral in interacting QFTs.

Rivasseau book "from perturbative to constructive renormalization" and Glimm & Jaffe "quantum physics a functional integral point of view" are also great. They use a lot Cluster Expansion methods.

There are also a bit of nice papers by Grosse with Whulkenhaar (first non-trivial QFT construction in 4D), Abdesselam ("Rigorous quantum field theory functional integrals over the p-adics"), Chaterjee ("A probabilistic mechanism for quark confinement").

For rigorous perturbative QFT, the most-complete thing that I know is Herscovich's book "Renormalization in Quantum Field Theory (after R. Borcherds)" that maybe is even harder than the other nonperturbative works cited here. Also a paper of Viet Dang "Renormalization of Quantum Field Theory on Riemannian manifolds". It uses a lot of REALLY HARD MATHS here.

pd: Sergio Albeverio also works with non-standard analysis, that are pretty cute.