word choice - Which article to use: 'the' or 'a'

1. 
   

   We adopt the approach under which the ...

   
   


2. 
   

   We adopt an approach under which the ...

   
   


3. 
   
   

   We consider the scenario where ...

   
   


4. 
   

   We consider a scenario where ...

   
   

Suppose that I say nothing about the approach or the scenario before there corresponding sentences. However, suppose that the scenario is known in the literature.

I am a bit confused about the article usage here since I am explaining/clarifying things about the approach (resp. scenario) after under which (resp. after where)

Question: (a) which one between 1 and 2 is correct ? (b) which one between 3 and 4 is correct ?

Answer

It seems you understand that the definite article is used when you are talking about a specific case, and the indefinite article used when talking about a non-specific or general case. Here, however, since you're introducing something new, both mean almost exactly the same thing. There is only a slight different in nuance, implying specific or non-specific, but the essential meaning is unchanged either way.

Depending on context, "the" is slightly more confident and assertive, than "a/an". "The" is more common in things like advertising because it implies that the service or product is the specific one you will need:

We have created the website that will answer all your English-language questions

We have created a website that will answer all your English-language questions.

It's not always appropriate to be too assertive, especially with things like scientific or technical papers where many approaches are possible. In your example, "the" is appropriate to draw focus to this one, specific approach. "A" suggests that you might talk about other approaches later, or that you want to imply the approach is generic. Again, both mean much the same thing and any additional meaning will depend on context.

If you add any modifiers to the noun, usually one or the other will work better, but this varies:

We determined the best approach where ...

We defined a new approach where ...