My new passport won't have Schengen visa?

Question to long-term non-EU residents of Belgium: My US passport expires later in 2012, so I'll have to have it replaced soon. I have a 5-year permanent residence card for Belgium (and I don't need to worry about extending my card for a couple more years).

Obviously, my new US passport won't have my original "D" Schengen visa for Belgium in it. Is it fine for me to continue to reside in Belgium (considering I have a Belgian perm. residence card) without also having a valid Schengen visa in my passport?

I rather assume that the Schengen visa was only needed for me to originally enter Belgium to live, and the need for it is now superceded by my residence card. I don't need to get a replacement Schengen visa once I have a new (visa-less) US passport do I?

I just want to avoid any problems flying in or out of Europe, or eventually renewing my Belgian residence card, if it turns out that when replacing your passport, you need to get an updated Schengen visa for it also.