  We find supersymmetric partners of a family of self-adjoint operators which
are self-adjoint extensions of the differential operator $-d^2/dx^2$ on
$L^2[-a,a]$, $a>0$, that is, the one dimensional infinite square well. First of
all, we classify these self-adjoint extensions in terms of several choices of
the parameters determining each of the extensions. There are essentially two
big groups of extensions. In one, the ground state has strictly positive
energy. On the other, either the ground state has zero or negative energy. In
the present paper, we show that each of the extensions belonging to the first
group (energy of ground state strictly positive) has an infinite sequence of
supersymmetric partners, such that the $\ell$-th order partner differs in one
energy level from both the $(\ell-1)$-th and the $(\ell+1)$-th order partners.
In general, the eigenvalues for each of the self-adjoint extensions of
$-d^2/dx^2$ come from a transcendental equation and are all infinite. For the
case under our study, we determine the eigenvalues, which are also infinite,
{all the extensions have a purely discrete spectrum,} and their respective
eigenfunctions for all of its $\ell$-th supersymmetric partners of each
extension.
