doi:10.1093/imrn/rnm148 Complex Zeros of Eigenfunctions of 1D Schrödinger Operators

In this article we study the semiclassical distribution of the complex zeros of the eigen-functions of the 1D Schrödinger operators for the class of real polynomial potentials of even degree, with fixed energy level, E. We show that as hn → 0 the zeros tend to concen-trate on the union of some level curves (S(zm, z)) = cm where S(zm, z) = ∫ z zm V(t) − E dt is the complex action, and zm is a turning point. We also calculate these curves for some symmetric and nonsymmetric one-well and double-well potentials. The example of the nonsymmetric double-well potential shows that we can obtain different pictures of com-plex zeros for different subsequences of hn.