Where do you apply for a license to operate on the LTE radio spectrum for amateur radio in the USA? Can this amateur license be used to operate a cell phone independent of a carrier?
An amateur radio license only grants privileges on the amateur bands. You would need a different type of license to use other bands. The cell companies pay a premium for the license for their spectrum, and they are almost certainly not interested in relicensing it (if they even could).
Moreover, LTE is just a protocol, so there is no dedicated spectrum for LTE in general. Each company that uses LTE uses their own spectrum, which differs per carrier.
So no, an amateur radio license does not let you run your own cell phone independently of carriers. It grants you privileges to use certain modes (voice, Morse code, etc) on well defined spectrum, but does not grant privileges for what you're asking about.