@spidergal,
I would definitely talk with HR. Do you have a cell phone with a camera?
If so then, while he stares at you, I would point my phone and
conspicuously take his picture. If he happens to actually walk
up to you and ask him why you did that tell him the truth. Say
"I'm taking your picture because I don't know your name and I
need a way to identify you to HR." When he asks why you are
talking to HR about him tell the truth again. "Your blatant staring
At me all through every lunch is rude and unnerving and so I am
reporting you." This could get you his name and at least a
reason from him about why he is doing this.