[Galaxy Note 3 Q&A, Help & Troubleshooting] StarTAC Fan:
Get an unlocked N9005, not a USA Note 3 Variant

In summary, you cannot use T-Mobile or any American variant (N900x) of the Galaxy Note 3 for what you want. You need to get one of the "International Model" (N9005x).

You are concerned ONLY about whether the phone will work on LTE networks. In particular, you are interested in Softbank of Japan and Vodofone of Italy. Have to looked at this? http://en.wikipedia.org/wiki/List_of_LTE_networks

You may already be aware, but according to the compilation in Wikipedia by the user community, Softbank currently uses BOTH 2100 MHz (Band 1) and 1700 MHz (Band 4). They supposedly will add support for 900 MHz (Band 8) this year. However, you may be able to use Softbank's 1700 MHz (Band 4) with the T-Mobile version. No American GN3 will support Vodofone Italy's 1800 MHz (Band 3) and 800 MHz (Band 20). No American Galaxy Note 3 will support ALL of those even if you go through the bother of rooting and flashing new modem files - there is just no physical, electronic support. Neither can the International Model be made to support American frequencies in their entirety.

http://xdaforums.com/showthread.php?p=45917884#post45917884

Here I discuss the "International Model" of the GN3 vs "North American". The International Model (GN9005x), as you may be aware, supports 800/850/900/1800/2100/2600 MHz. It will work on Softbank because of 900 and 2100 MHz support. It will work on Vodofone Italy because of 800 and 1800 MHz support. But none of those LTE frequencies are used in North America (except for 2600 MHz in Canada) if you plan to travel there. (WCDMA support outside of Japan and Italy is a different matter if you are satisfied with that).

http://xdaforums.com/showthread.php?p=46031869#post46031869

http://support.t-mobile.com/docs/DOC-9056

You made specific mention of 1700 MHz - not sure if that is for use with Softbank or many North and South American carriers such as T-Mobile. If you specifically require LTE 1700 for T-Mobile, you're out luck. At best, the T-Mobile version supports LTE 2100 and 2600 MHz which are used in many countries around the world but not in Italy. The 2100 MHz can be used for Softbank. (I believe Softbank uses 1700 MHz for both up/down, and again 2100 MHz for both up/down. I am not sure if the T-Mobile version is fixed at 1700/2100 up/down or can interchangeably work with 1700 up/down and 2100 up/down. It's probably interchangeable.)

If LTE 1700 MHz is for Softbank, you may not need it at all. Their 2100 MHz and upcoming 900 MHz coverage area will likely overlapped with 1700 coverage areas. Often the spectrums are are stacked in order to handle traffic volume. Thus again the GN9005x will be suffice.

So the general advice here is to get a GN9005x, not a USA Notes 3 variant (as is the title of this thread).

Then, which GN9005x should you get? I don't know what are all frequencies supported by the GN3 sold by Softbank or Vodofone Italy. You'll have to ask them - I don't know if either one may be fully compatible with the other. I would imagine Softbank and Vodofone sells phones locked to their network but allow some roaming capability.

On the other hand, you could get an unlocked phone which will remove doubts. In that case, any N9005x will be work. The MPN after the N9005 refers to the carrier or country (eg. N9005ZKEXSI for Arab Emerites). That's why Samsung seems to have so many different models of the N9005's. The carrier enables or disables certain things in firmware but electronically they are identical. Unlocking removes certain limitations.

Many people here at XDA seem to prefer buying new pre-unlocked phones from Negrielectronics including the N9005. (I don't have any affiliation with them.) . There are of course private sellers who may have unlocked ones. You may be able to find a locked N9005 including those sold by Softbank or Vodofone and unlock it yourself using various online services but that will generally cost an additional $30. Other factors for you may be shipping cost and import charges if you have one sent from outside of your country where ever that may be.

ADD: Why your Softbank phone will not work in the US.

I didn't notice this before, under your name it says you are in Tokyo. So I guess your home carrier is Softbank.

Softbank LTE:

- 900 MHz Band 8 <The International Galaxy Note 3 will support this.

- 1500 MHz Band 11

- 2100 MHz Band 1 <Supported by both the American and International Galaxy Note 3.

EMOBILE (Softbank) LTE:

- 700 MHz Band 28 (note this is not the same as 700 MHz Band 17 and 29 with AT&T nor Verizon Band 13 in the US)

- 1700 MHz Band 9 (note this is not the same as 1700 MHz Band 4 used by T-Mobile)

The reason why the same frequency is not necessarily the same band is that frequency is not precise. It's not exactly 700.0 MHz for example. It's actually a range. So "700 Mhz" actually refers to anything between about 700 MHz to about 800 MHz. Different carriers buy a right to use a portion of that from the government. So for example, AT&T's 700 MHz refers to somwhere between 700 MHz to 750 MHz. This is the sometimes called the "lower 700". Verizon's 700 MHz refers to somewhere between 750 MHz to 800 MHz. This is sometimes referred to as "upper 700". These are further divided into blocks. So for example, between 700 and 750 MHz, there are 8 blocks. AT&T owns 6 out of those 8 (other small carriers share 2 blocks). AT&T currently use 4 of them (ie. 2 pairs of blocks) and have the other 2 on reserve (ie. 1 pair). They work in pairs because there has to be a block used for signals from phone to tower and another for tower to phone. The 2 pairs that AT&T uses in the 700 to 750 range are called "lower 700b" and "lower 700c" and they occupy 25 out of 50 MHz in that range. Together, these are called Band 17. The 1 pair that Verizon uses also occupy 25 Mhz (it's larger than AT&T's 2 smaller ones) in the 750 to 800 range. It is called "upper 700c" and this is also referred to as Band 13. Band 17 and 13 are not compatible. Similarly, Softbank's Band 28 is incompatible with both AT&T and Verizon...even though all 3 bands in the 700 to 800 MHz range!

To visualize what I just said, look at: http://www.phonescoop.com/articles/article.php?a=187&p=232

What this means for you is that a phone designed for Softbank's 700 MHz Band 28 probably (I don't know for sure if their phone's modem can adjust) cannot be used with AT&T nor Verizon. And their phone designed for their 1700 MHz Band 9 probably cannot be used on T-Mobile's 1700 MHz Band 4. Having the most number of bands, LTE is the most fragmented and confusing standard ever.

This does not change the original recommendation of getting an N9005. It's the only Galaxy Note 3 that will work on both Softbank and Vodofone Italy. Make sure if you buy it from Softbank, make sure it will allow roaming on Vodofone Italy. If you buy an unlocked one, it doesn't matter.