First of all, let me say, great question and great responses from everyone! 

I'm gonna go through your awesome post and post quick questions for quick consideration :), which hopefully will help! I am confident that if you read your own post again after my questions, your doubts will clear up! 

Q1) When she is looking for a seat, are there open seats for her to sit ***further away*** from you? Or is she always sitting as you said "near me"? In your title, you even said "next to me"! If this is a pattern, that is awesome! 

Q2) Could it be that is she ***intentionally sitting*** close to you? If no other seats are open, then that is a different story. 

Q3) Does she tend to sit in a table ***close to you*** and ***sit on the side facing you***? Where she places her stuff on the side closer to you, ***so she could possibly see you*?** Or does she sit facing away from you? 


You said some great things in your post that definitely show a pattern of interest to a point where you yourself said "feel the vibes". You said you have both "seen each other", even **"smiled at each other"** at each other", and to top it all off **"for a few weeks now"**!! That is great! So that being said,  

Q4) Is she mostly smiling ***at you***? Or does she tend to smile at everyone else in her sight as a nice gesture? She might be a friendly person in general. 

True, wearing headphones does indicate that she might not want to be bothered. But, here comes the next, and possibly the most important questions. 

Q5) With her headphones always on, why would she go so far as to ***smile at you*** if she doesn't want to be bothered? Place yourself in her shoes. She is busy. You are busy. If ***you*** don't want to be bothered, why even sit close to people and not in more secluded areas? ***Why smile at people?*** If anyone is busy at a cafe, wouldn't they prefer to focus on getting work done, not spend social time there, and do it with their friends at a different place smiling then? Wouldn't you? 

Q6) ***Could it be that she wears her headphones always regardless of the situation? Both when she studying, and when she is facebooking, msging her friends? Walking in the cafe, and out? Are they always on meaning that the headphones are on regardless of the situation?***

Q7) Why would someone go out of their way to ***smile at you for weeks now***? ***sit next to you for weeks now***? and most importantly ***respond to your smiles at her for weeks now***?  

And the last one question 
***Could her smiles be her way of signaling to you to start a convo that she is shy of starting herself?*** ***Could she be wearing the headphones because she is shy?"*** Just because someone is attractive doesn't mean that they are confident conversation starters. 

So, my advice is, like many others, leave her a note. Or, be a little more clever and leave her something like the picture below in a message screen or a note taking app so she can respond, and see how she acts. I did this in a quiet library with a lot of people, and it started the conversation, plus she said it was really cute. Nobody else cared. 
***It also gave her control to start or stop the conversation.*** 

[![enter image description here][1]][1]

 


  


  [1]: https://i.sstatic.net/pm4Zs.jpg