Let D and H be the self-adjoint, one-dimensional
Dirac and Schr\"odinger operators in L2(R;C2) and
L2(R;C) respectively. It is well known that, in absence
of an external potential, the two operators are related through the equality
D2=(H+41β)1. We show that such a
kind of relation also holds in the case of n-point singular perturbations:
given any self-adjoint realization D of the formal sum
D+βk=1nβΞ³kβΞ΄ykββ, we explicitly determine
the self-adjoint realization H of H1+βk=1nβ(Ξ±kβΞ΄ykββ+Ξ²kβΞ΄ykββ²β) such that
D2=H+41β. The
found correspondence preserves the subclasses of self-adjoint realizations
corresponding to both the local and the separating boundary conditions. The
case on nonlocal boundary conditions allows the study of the relation D2=H+41β for quantum graphs with (at most) two ends; in
particular, the square of the extension corresponding to Kirchhoff-type
boundary conditions for the Dirac operator on the graph gives the direct sum of
two Schr\"odinger operators on the same graph, one with the usual Kirchhoff
boundary conditions and the other with a sort of reversed Kirchhoff ones