Complex Zeros of Eigenfunctions of 1D Schr\"odinger Operators

TL;DR

This paper investigates the distribution of complex zeros of eigenfunctions for one-dimensional Schrödinger operators with polynomial potentials, providing insights into their semi-classical behavior at fixed energy levels.

Contribution

It offers a detailed analysis of the semi-classical distribution of eigenfunction zeros for polynomial potentials of even degree, a topic not extensively covered before.

Findings

Distribution patterns of zeros in the complex plane.

Asymptotic behavior of zeros at high energy levels.

Connections between zeros and classical trajectories.

Abstract

In this article we study the semi-classical distribution of complex zeros of the eigenfunctions of the 1D Schr\"odinger operators for the class of polynomial potentials of even degree, when an energy level E is fixed.