During the library preparation it's possible to conserved Ampure beads during the different enzymatic reaction (it's the with beads strategy for improving yield, Fisher et al. Genome Biology 2011). Nevertheless, after double size selection, the library is always eluted from Ampure before PCR amplification. I want to know if it is possible to carry out a PCR directly on library not eluated from Ampure beads after size selection ?
it depends on which PCR polymerase you're using. When doing some previous methods development, I tested about half a dozen polymerases amplifying either directly on the beads or after elution. Some polymerases worked equally well under either condition, while others were completely inhibited by the presence of beads. However, elution from the beads is essentially quantitative (I've measured >99% if you're good at getting every last microliter), so it's probably a safer bet not to risk your samples unless you have a strong incentive not to elute.
Provided there isn't lots of PEG (the "magic" ingredient in Ampure/SPRI beads) or ethanol remaining in your elution volume the DNA will lose its affinity to the beads. Plus, in a PCR there are many high-temperature steps which would only help "elute" DNA off the beads anyway.
Now, whether the presence of the beads inhibits a particular polymerase is another matter. I've seen it, though I don't recall which polymerase. However, I know for a fact that it does work with Platinum Taq polymerase (Invitrogen cat# 10966034). The protocol in the following publication says to elute but not remove the beads for a subsequent PCR step (though you would miss it if you weren't paying very close attention). And for me, that part of the protocol works as written. https://www.nature.com/articles/nbt.3117