Non-Supersymmetric Vacua and Self-Adjoint Extensions

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.