NO TRAFFIC ADVISORY GIVEN WHEN IN CONTACT WITH APCH CTLR AND SQUAWKING ASSIGNED CODE.

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WHILE AN A VFR FLIGHT FROM TOA TO WJF IN LANCASTER; CA I EXPERIENCED A NEAR MISS WITH A LTT JUST NORTH OF BURBANK. AFTER EXITING THE LAX CORRIDOR I CONTACTED BURBANK APPROACH NE OF LAX WHILE STILL UNDER THE TCA. I WAS ASSIGNED A TRANSPONDER CODE ALTHOUGH THE BURBANK CTLR EXPRESSED SOME QUESTION ABOUT WHICH CTLRS SECTOR I WAS IN. HE WAS VERY BUSY WITH NO PAUSE IN HIS XMISSION TO OTHER TFC AND HE DID NOT VERIFY RADAR CONTACT WITH ME. I CLBED TO 7500' AND WAS IN LEVEL FLT WHEN SUDDENLY I WAS NOSE TO NOSE WITH A LTT AT ELEVEN THIRTY TO TWELVE O'CLOCK APPROXIMATELY 50 TO 100' AWAY JUST SLIGHTLY BELOW ME. I INSTINCTIVELY PULLED BACK ON THE YOKE AND ABRUPTLY CLIMBED TO AVOID COLLIDING WITH HIM. I ASKED BURBANK IF THEY KNEW ABOUT OUR NEAR COLLISION. THEY GAVE ME A NUMBER TO CALL AFTER I LANDED AT WHICH TIME THEY STATED THAT EVEN THOUGH THEY HAD GIVEN ME A TRANSPONDER CODE THEY HAD NOT VERIFIED MY POSITION AND THEN LOST ME ON RADAR BECAUSE I POSED OVER THEIR ANTENNA. THEY DID POINT ME OUT TO THE LTT AS UNCONFIRMED AT 7500'. THE LTT WAS DESCENDING THROUGH 7500. I ASKED WHY THE CTLR DIDN'T GIVE ME ANOTHER CTLR WHEN I FIRST CONTACTED HIM IF HE WAS TOO BUSY OR WORKING A DIFFERENT SECTOR. I WAS TOLD THAT THE OTHER CTLR (TO WHOM I WAS EVENTUALLY HANDED OFF TOO AFTER THE NEAR MISS) WAS JUST AS BUSY. EVEN THOUGH I PASSED OVER THE ANTENNA I DON'T UNDERSTAND WHY THE COMPUTER DIDN'T PROJECT MY CONTINUED COURSE AND WHY THE COLLISION AVOIDANCE SYSTEM DIDN'T ALERT THE CTLR THAT BASED ON OUR MODE C INFORMATION AND COURSE WE WERE GOING TO COLLIDE. IT ALSO APPEARS THAT THE CTLRS WERE TOO BUSY. IF ONLY TWO CTLRS WORK THE BURBANK ARSA; THEY ARE TOO SHORT STAFFED.