Learjet 60 Captain reported a traffic conflict while turning from base leg to final at a non-towered airport. As the Captain started the turn TCAS became active. The TCAS then turned to a TA and the offending traffic was spotted. A small jet flew 500 feet overhead and continued to land. Captain reported no communications from the small jet during the incident.

Narrative

We were on an IFR flight plan and had been handed off for the visual approach to the non-towered airport (ZZZ). We were #1 to the field with 3 jets to follow us. We were listening/ talking to both CTAF and Center as we maneuvered upwind/overhead the runway; at pattern altitude; to make right traffic for Runway XX; as we entered the right base for Runway XX I relayed our 'cancel IFR' through the airplane behind us; as we continued the turn to final; I made my last call of 'short final and that is when we got the first traffic return on our TCAS; we had not observed another unknown traffic up to this point; followed rapidly by it turning yellow and advising 'TRAFFIC TRAFFIC' I quickly scanned left and saw the Aircraft X turning in on top of us; we had 700 ft. separation at this point; he got to within 500 ft. by the time he had past above; us and clear. We never heard him on center frequency or on the local CTAF (XXX.X) and in fact never heard or got a response from him until after he landed and called taxiing in; not clear of the runway. This occurrence could have been completely avoided if the pilot of Aircraft X had used standard pilotage. He did not communicate with anyone in the area and he did not use the standard entry for the pattern; which all airplanes were using; and was just carelessly making his way through the sky. When we departed we observed him leave VFR; no clearance; out through the valley and once again; talking to nobody. While what he did was legal; it was highly careless for the busy; tight; mountainous airport! He choose to operate VFR so that he could be more expeditious and cut in front of everyone. His lack of patience and procedure almost ended in tragedy!"