He killed Alucard BY commiting suicide. It's complicated.
Basically, he killed himself, but made sure his body and blood fell into a giant river of blood that Alucard was consuming. Therefore, Alucard accidentally merged with Schrodinger. Specifically, Schrodinger merged with Alucard plus every soul Alucard ever consumed.
In order to exist, Schro has to recognize himself, and he could not do this if he's a million souls at once. Therefore he ceases to exist, as well as all the souls he combined with, including Alucard.
tl;dr HE HAXED ALUCARD AND DEVIDED BY ZERO