For the most part Amtrak runs on rails owned and maintained by private companies. Plenty of exceptions - especially the Northeast Corridor - but the vast majority of track is outside government control.
I would say very few exceptions, other than the the Northeast Corridor. There are some tracks in a few other places (LA and Chicago, mainly) owned by commuter rail authorities, but so far as I know, the Northeast Corridor is the only real intercity line not owned by a private company.
However given how congress is trying to defund everything, the infrastructure this runs over is probably dangerous as hell.