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.
I am a UK licensed Radio Amateur with some 39 years experience. I also design and build LTE cellular systems for a living.
The best suited Amateur Radio spectrum for LTE operation is the 2.3 GHz band - this can easily accommodate a LTE-TDD (single frequency) system operating in LTE Band 40.
Note that both FCC (and European) regulations for Amateur Radio do not allow the transmission of ENCRYPTED communications - and that LTE is designed with encryption integrated into the system; however, this can be got around by using PUBLICALLY KNOWN keys, i.e. use keys consisting of all'0's.
I can see Amateur LTE being particularly useful in emergency/disaster situations as a means of communicating with members of the public in distress - although there are more regulatory issues involved with licensed amateurs communicating with unlicensed operators.
Alternatively, as a previous poster has indicated - it is possible to obtain an experimental license from the FCC for LTE operation. Note - it is not possible to run a commercial service on experimental spectrum.
Actually LTE standard could be adapted for Amateur for certain bands Amateur radio has some packet switching radios where you can connect to the internet but like Ham radio it is so slow. If Ham radio wants to survive into the 21st century we need to get it away from these old gray haired men and give it new life. Get the internet working at very high speeds. LTE advanced can go as high as 3GIG Amateur has 1.2GHZ band and 440 we could use those bands and implement LTE. You can't sell the service but provide it for free to other Ham radio operators.