In this paper, we identify the deeper structure that produces shape invariance. We show that shape invariance appears when a quantum mechanical model is invariant under a centrally extended superalgebra. Also we show that in Φ4 and sine–Gordon theories, shape invariance arises from the Bogomol’nyi–Prasad–Sommerfeld (BPS) structure associated with shift operator. The shift operator in two models ensures that there is a one-to-one correspondence between the energy levels and energies of the BPS saturating.
