Resolvent formalism in quantum mechanics

[Konte]

Hi everybody,

While reading some quantum mechanics book, I met the resolvent formalism which is presented as more powerful than the pertubative approach. For a system with a hamiltonian $H = H_0 + H_{int}$, when the interaction part $H_{int}$ is no more a pertubation but rather having the same importance as $H_0$, the perturbation theory is obviously no more appropriate to solve the problem.
My question:
Is the resolvent formalism can solve this specific case and viewed as the appropriate approach for?

Thanks

 

[A. Neumaier]

The resolvent formalism which is presented as more powerful than the pertubative approach. For a system with a hamiltonian $H = H_0 + H_{int}$, when the interaction part $H_{int}$ is no more a pertubation but rather having the same importance as $H_0$, the perturbation theory is obviously no more appropriate to solve the problem.
My question:
Is the resolvent formalism can solve this specific case and viewed as the appropriate approach for?

It depends on what you understand by ''solving'' the problem. The resolvent formalism is exact and nonperturbative, but to get numerical results out of it one needs to make approximations at some point. These approximations may or may not be in terms of perturbation theory.

 

[Konte]

The resolvent formalsism is exact and nonperturbative, but to get numerical results out of it one needs to make at some point approximations. These approximations may or may not be in terms of perturbation theory.

What I wanted is to reassure myself about the scop of the resolvent formalism in comparison to other approach.
Now, through your answer, I understand that resolvent formalism is not even an approximation but an exact method.
I put aside all about numerical results, I am only interested in the nature and the formal aspect of different methods.

Thanks

 

[Konte]

The resolvent formalism is exact and nonperturbative.

I am back for some precisions. What do you mean by "nonperturbative"? Can I understand it by: the resolvent formalism always works for all $H_{int}$ even if it is as big and strong as $H_0$?

Thanks

 

[A. Neumaier]

I am back for some precisions. What do you mean by "nonperturbative"? Can I understand it by: the resolvent formalism always works for all #H_{int}# even if it is as big and strong as #H_0#?

Yes, if we refer to the same by ''the resolvent formalism''. Since nowhere any approximation is made, strength nowhere enters. If you want more details, provide the formulas relevant for a deeper discussion.

 

[Konte]

Yes, if we refer to the same by ''the resolvent formalism''.

In your reply, you seem like hinting that there is another formalis of resolvent or am I misunderstanding?

If you want more details, provide the formulas relevant for a deeper discussion.

Thanks for your answer. In reality, I am discovering this formalism of resolvent right now, while reading the Cohen Tannoudji books (interaction process between photons and atom- french version). It is amazing for me that formalism stronger than perturbation approach exists. It seems like difficult and complex but, be that as it may, I want to understand deeply and master it. Could you advice me some lectures or books that can help me in this sense?

 

[A. Neumaier]

It is amazing for me that formalism stronger than perturbation approach exists.

Perturbation theory is the weakest of all approaches. Its the stuff that one begins with because it is so easy to use, as long as it works.

I can only repeat:

If you want more details, provide the formulas relevant for a deeper discussion.

 

[Konte]

If you want more details, provide the formulas relevant for a deeper discussion.

Ok. As you know, I'm just starting to read this book and learn slowly the formalism, I may need time. But I will be back soon to discuss deeply about. Sure.

Thanks