Analytic Continuation of Eigenvalues of a Quartic Oscillator - Eremenko, Alexandre; Gabrielov, Andrei (Communications in mathematical physics, 2009)
We consider the Schrödinger operator on the real line with even quartic potential x 4 + α x 2 and study analytic continuation of eigenvalues, as functions of parameter α . 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 α -plane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the α -plane.