Air carrier inbound to JFK experienced TCAS RA with VFR issued traffic in another Controller's airspace; ATC indicating he was unsure of the VFR aircraft's intended track/altitude.

Narrative

Our flight was given vectors from the Kenebunk 4 Arrival for the VOR 13L at JFK. Our controller descended us to an altitude of 7000' on a heading of 240 deg. Then the controller told us of VFR traffic 12 o'clock; 5 miles; at 7500' moving northeast; but he said he did not know the traffic's intentions. Both crew were starting to wonder if the controller really knew what this traffic was doing and who was controlling him. We did not call back that we could see the traffic and the controller again called out this VFR traffic now 12 o'clock; 3 miles at 7500'. Both crew picked up the traffic visually and on the TCAS. We did not hear any further communications from ATC until after the event was over. We could start to see on the TCAS the traffic was descending now at 2nm and 300' above us. I stated to the First Officer to prepare for a RA and momentarily after we received a RA warning to descend. The VSI guidance showed a 2000 fpm descent required. I manually descended and the traffic passed within 500-600' vertically and 1/4nm horizontal. The TCAS advised a level off at 6300'msl with a 'monitor vertical speed' call-out. After we heard clear of conflict we returned to 7000'msl and advised ATC of the conflict. ATC acknowledged but did not add anything further. I asked for a phone number to call ATC later and we continued the approach to landing normally. I did call the ATC number and talked to that specific Controller. He explained that the VFR traffic we came close to was under positive control with the Departure Controller and he; the Approach Controller; did not have any contact with that target nor the Controller who was handling him. I asked if his workload prevented him from vectoring us away from the VFR traffic before the TCAS RA and he said he did not know the traffic was descending and he thought the target was outside the NYC Class B airspace. Even though VFR traffic are handled under different frequencies; if they are being vectored in the same area of IFR arrival/departure traffic then some cross communication between controllers should be necessary. If the traffic is outside their airspace; then we should be vectored away.