QUESTION about buying an unlocked International Note 2 - will it work on EITHER T-Mobile or AT&T?

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http://forums.androidcentral.com – Hi, Ever since owning a mobile phone, I have been on either Sprint, or now, Verizon, so I have a noob question about GSM in general. I want to purchase a Note 2 outright, and then have the option to use either T-Mobile or AT&T on a month-to-month basis. I live in Louisville, KY and hear mixed reviews of either carrier's coverage here, so I want to have that freedom to switch and test out each. On eBay, I see phones with specs listed below. Can someone please confirm whether this would work on either carrier? I see it says LTE in the title, and curious if that means it would work (HowTos)