  We consider the self-adjoint extensions (SAE) of the symmetric supercharges
and Hamiltonian for a model of SUSY Quantum Mechanics in $\mathbb{R}^+$ with a
singular superpotential. We show that only for two particular SAE, whose
domains are scale invariant, the algebra of N=2 SUSY is realized, one with
manifest SUSY and the other with spontaneously broken SUSY. Otherwise, only the
N=1 SUSY algebra is obtained, with spontaneously broken SUSY and non degenerate
energy spectrum.
