  Internal intervals spanned by finite ranges of a conformal coordinate $z$ and
terminating at a pair of singularities are a common feature of many string
compactifications with broken supersymmetry. The squared masses emerging in
lower-dimensional Minkowski spaces are then eigenvalues of Schr\"odinger-like
operators, whose potentials have double poles at the ends of the intervals. For
one-component systems, the possible self-adjoint extensions of Schr\"odinger
operators are described by points in $AdS_3 \times S^1$, and those
corresponding to independent boundary conditions at the ends of the intervals
by points on the boundary of $AdS_3$. The perturbative stability of
compactifications to Minkowski space time depends, in general, on these choices
of self-adjoint extensions. We apply this setup to the orientifold vacua driven
by the ``tadpole potential'' $V=T \ e^{\,\frac{3}{2}\,\phi}$ and find, in nine
dimensions, a massive scalar spectrum, a unique choice of boundary conditions
with stable tensor modes and a massless graviton, and a wide range of choices
leading to massless and/or massive vector modes.
