High energy eigenfunctions of one-dimensional Schrodinger operators with polynomial potentials

TL;DR

This paper investigates the asymptotic distribution of zeros of scaled eigenfunctions for a class of one-dimensional Schrödinger operators with polynomial potentials, revealing dependence on potential degree and boundary conditions.

Contribution

It establishes the limit distribution of zeros in the complex plane for eigenfunctions of polynomial potential Schrödinger operators as eigenvalues grow large.

Findings

Zeros of eigenfunctions have a well-defined limit distribution in the complex plane.

The limit distribution depends solely on the polynomial degree and boundary conditions.

Results apply to both Hermitian and PT-symmetric operators.

Abstract

For a class of one-dimensional Schrodinger operators with polynomial potentials that includes Hermitian and PT-symmetric operators, we show that the zeros of scaled eigenfunctions have a limit disctibution in the complex plane as the eigenvalues tend to infinity. This limit distribution depends only on the degree of potential and on the boundary conditions.