The Complex Airy Operator as Explicitly Solvable PT-symmetrical Model

TL;DR

This paper analyzes a PT-symmetric Sturm--Liouville operator with an explicit potential, revealing eigenvalue behaviors across a range of the physical parameter using Airy functions.

Contribution

It provides an explicit description of eigenvalue dynamics for a PT-symmetric operator with a linear complex potential, identifying critical parameter values.

Findings

Eigenvalue behavior changes with the parameter 

Critical  values are expressed via Airy functions

Eigenvalues exhibit a notable phenomenon as  varies

Abstract

We study the Sturm--Liouville operator $$ T(\varepsilon)y=-\frac{1}{\varepsilon}y''+ p(x)y, $$ with concrete $\mathcal{PT}$-- symmetric potential $p(x) = ix$ and Dirichlet boundary conditions on the segment $[-1,1]$. Here $\varepsilon \in (0, \infty)$ is a physical parameter. We explicitly describe a beautiful phenomenon of the eigenvalue behavior when $\varepsilon$ changes from $0$ to $\infty$. All the critical values of $\varepsilon$ which determine the eigenvalue dynamics, are found in terms of the special Airy functions.