Is it possible that a reformulation of the creation or annihilation operator would enable us to bound the S-matrix?

Question

Alright I'm not sure if this question is too speculative but here goes: 

So in my graduate days I remember wishing I could put some kind of bounds on the S-matrix. I came to understanding that one of the problems was there was no other way to talk about the creation and annihilation operator in than the standard method in bra-ket notation. Due to this I decided to focus on creating my own formulation of them in Quantum Mechanics. I did (partially) succeed: https://mathoverflow.net/questions/301699/can-one-calculate-the-following-operator
 

One can define a creation operator:

$$A^\dagger | n \rangle = | n+1 \rangle$$

In fact, 

$$A^\dagger = |2  \rangle \langle 1 | + |3  \rangle \langle 2| + |4  \rangle \langle 3 | +  \dots$$

The quantum mechanical creation operator is $\hat A^{\dagger} \leq \hat a^{\dagger}$

Taken from the answer (but edited due notation consistency error):

... we interpret $|x\rangle \langle y|$ as a rank 1 operator. In particular, $\hat{n}$ is a coisometry which takes the orthonormal set $\{|n\rangle, |2n\rangle, |3n\rangle, \ldots\}$ to the standard basis $\{|1\rangle, |2\rangle, |3\rangle, \ldots\}$. The "rational operator" $\frac{\hat{m}}{n}$ takes the basis vector $|kn\rangle$ to $|km\rangle$ when $kn$ is an integer...

Now we make our reformulation:

$$(\hat 1 - A^\dagger)^{-1} = ( \sum_{0 < R \leq 1} \hat R)^\dagger$$
where $( \sum_{0 < R \leq 1} \hat R)$ represents the sum of all rational operators whose corresponding rational number is less than or equal to $1$.

Assuming the math somehow works itself out (?) Would a reformulation of the creation and annihilation operator help me in the endeavour of "bounding the S-matrix"? Have other people tried similar games for the similar purposes? 

Comments

Your operator does not have the properties needed to reproduce the standard formulas for one harmonic oscillator, let alone for a scattering problem. The operators introduced in textbooks are chosen not for reasons of simplicity but such that they give agreement with experiment! Of course you can play games by creating your own quantum mechanics but this is no physics but a pastime.