Empty (2011)

Koban Kato is a quiet high school student whose presence is hardly ever noticed. Being unable to determine his own existence, Koban gains the ability of teleportation. Koban despised this power that proved no one cared about him. One day, Koban unconsciously teleports to a deserted place where he meets a lonely yet beautiful girl named Sheena. Sheena becomes attracted to Koban's ability to teleport and decides to accompany him. Koban uses his special powers for Sheena, but the mysterious incidents caused by their actions start rumors to spread in town. Members of the Community Development Committee, who are concerned of their town's severe depopulation, hear about these rumors and plan to utilize this to their own advantage.