The Complex Airy Operator as Explicitly Solvable PT-symmetrical Model

Abstract

We study the Sturm--Liouville operator $$T(\epsilon )y=-\frac{1}{\epsilon }{y}^{\prime \prime }+p(x)y,$$ with concrete $\mathcal{P}\mathcal{T}$-- symmetric potential $p(x)=ix$ and Dirichlet boundary conditions on the segment $[-1,1]$. Here $\epsilon \in (0,\infty )$ is a physical parameter. We explicitly describe a beautiful phenomenon of the eigenvalue behavior when $\epsilon$ changes from $0$ to $\infty$. All the critical values of $\epsilon$ which determine the eigenvalue dynamics, are found in terms of the special Airy functions.