  We consider the Schrodinger operator on the real line with even quartic
potential and study analytic continuation of eigenvalues, as functions of the
coefficient of the potential. We prove several properties of this analytic
continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues
are given by branches of two multi-valued analytic functions, one for even
eigenfunctions and one for odd ones. 2. The only singularities of these
multi-valued functions in the complex plane are algebraic ramification points,
and there are only finitely many singularities over each compact subset of the
plane.
