Complex Zeros of Eigenfunctions of 1D Schrödinger Operators

摘要

In this article we study the semiclassical distribution of the complex zeros of the eigenfunctions of the 1D Schrödinger operators for the class of real polynomial potentials of even degree, with fixed energy level, E. We show that as h_{n → 0} the zeros tend to concentrate on the union of some level curves ℜ (S (z_m , z)) = c_m where is the complex action, and z_m 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 complex zeros for different subsequences of h_n.