The Complex Airy Operator as Explicitly Solvable PT-symmetrical Model

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.