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 Schrodinger-like operators, whose potentials have double poles at the ends of the intervals. For one-component systems, the possible self-adjoint extensions of Schrodinger operators are described by points in AdS(3) x S-1, and those corresponding to independent boundary conditions at the ends of the intervals by points on the boundary of AdS3. 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(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.