This is a key question. In the 1494-95 French campaign I have read that one reason for his success was that he had a new siege train, which was much more mobile and powerful than before. That he managed to take a number of fortresses very quickly, including one that had recently withstood a 7 years siege. [Although checking the Wiki entry this seems to suggest a shortage of sieges].
I do know that in 1453 the Ottomans had to build their siege cannons in situ for the final capture of Constantinople. Also that this in turn depended on renegade Hungarian gun-makers.
I have asked this question when I have seen previous threads on this issue and never received any answer. Do the Ottomans have a mobile siege train capable of doing what the French could. If so they probably can take most of S Italy, in a similar way to the French. If they don't and would have to rely on storming, starving or building guns at each location, then they are unlikely to make deep or lasting progress in Italy. [In one way the relatively small area of S Italy, coupled with the rough terrain makes conquest more difficult than the broad plains of Hungary as there are bottlenecks.
I would say that if they could do it then the shock from this could well mobilise a coalition against them as its a significant increase in the threat to the Christian states of the western Med. That is a totally different question.
Steve
You are correct in saying that the Ottomans' standard practice in the second half of the 15th century was to cast siege bombards in place, and to break them up after the conclusion of the siege to recover the metal. The siege of Scutari is another example.
This probably was somehow changing towards the end of the century: at the siege of Lepanto in 1499 the siege guns were transported by the Ottoman fleet: this does not mean that the Ottomans had an easily movable train siege, and probably in the case of sieges far from the coastline they were still casting bombards in place.
At the first siege of Vienna (two generations after Otranto) the Ottomans brought up the siege guns from Belgrade, and it was a very slow going (IIRC it took 2 and half months to cover the 600 km from Belgrade to Vienna).
Charles VIII had a train siege which he took in his Italian campaign. However this happened 15 years after Otranto, and the late 1400s were an age of fast development in gunnery, both on land and at sea. However Chales VIII had allies in Italy, and a fleet providing support along the Thyrrenian coast, which might also be used to transport heavy guns.
You are also right when saying that there were just a few sieges in his campaign. OTOH, I'd point out that it was much more likely that invested cities would negotiate with a French king rather than with an Ottoman army and the Anjou party in Italy has also to be taken into account.
The attempted invasion of southern Italy was overall a very half-assed attempt: the Ottomans took easily Otranto, a small port at the very tip of the heel of the boot which was not defended. However the few probes toward Brindisi (which is a bigger city and port, but walled and defended) were easily repulsed by Neapolitan troops. A limited number of Ottoman troops was kept in Otranto for some time (and allegedly the Venetians were bringing them provisions), but the sultan lost interest very soon and decided (or was made to decide: there is this wild legend that the sultan did whatever he liked, but the influence of the different departments of the Ottoman government should not be underrated, as well as the fact that these different centres of power had also different interests and were riddled with jealousies) and went on the the unsuccessful siege of Rhodes.
The threat of an Ottoman invasion was taken seriously at the time, and the pope tried to put together a league including Aragon, Portugal and Naples, but the threat never materialised.
The choice of Otranto as the gateway for invasion was pretty poor in any case: the town was small as was the harbor, and its location was extremely periferic. The population also left en masse, after the massacre and all supplies had to be ferried across the sea.
There is also the small problem that Ottoman armies at the time relied on cavalry, and ferrying thousands of horse across the straits would have been a huge undertaking (and one made theoretically possible only by the peace recently signed with Venice in 1479). The logistics of supplying and feeding men and horses would IMHO be daunting, to say the least.
I also made the same point as you did about the difficult terrain: if an invasion has to succeed, the Ottoman troops have to negotiate the Appennines, and there was just a single road suitable: the via Trajana, from Brindisi to Benevento