No horn of singularities for the double well anharmonic oscillator

The author considers the one-dimensional quantum mechanical potential V = g2(x2 - 1/g2)2, which has pseudo-particle solutions. It has been suggested that the work of Bender and Wu, and Simon on the anharmonic oscillator shows that the energy levels for V, considered as an analytic function of g2, have a horn of singularities asymptotic to the real g2 axis. This implies that the eigenvalues of the hamiltonian could not be written as a Borel sum. It is shown here that there is no such proof, and indeed, this potential does not have a horn of singularities. Further, the known singularity-free region allows the possibility of Borel summability. (Auth.)