Common name: Spider Lily
Family: Amaryllidaceae
Genus: Lycoris
Species radiata
Given the right conditions, yes, they will keep spreading. Each bulb doesn't always bloom every year, though. One bulb may bloom one year, then not bloom the next year. A friend of ours joked once that she thought they moved underground on her as they were always popping up in different places. :)
If you need a home for your lycoris, there are lots of people on DG who would love to have them...myself included. I am now in OK where these bulbs aren't quite as prolific. :) You can get some good trades for them, too, as they're nearly always in demand. Good luck!
I counted my blossoms when I cut off the scapes for winter. I had 70+ in an 8' line.