Analytic problems arising in the study of $PT$-Symmetric Sturm–Liouville Operators

We consider $PT$-symmetric Sturm–Liouville operators
$$ T(\varepsilon) = -\frac 1\varepsilon\frac{d^2}{dx^2} + P(x), \quad \varepsilon >0, $$
in the space $L_2(-a,a)$, $0<a\leqslant \infty$, where the potential $P$ is subject to the condition $P(x)= -\overline{P(-x)}$. Potentials of this form are called $PT$-symmetric. The spectrum of these operators is symmetric with respect to the real axis and discrete, provided that the interval $(-a, a)$ is finite and the potential $P$ does not have high order singularities. It is also discrete when the interval coincides with the whole axis, provided some additional assumptions on the growth of $P$. The problem is to clarify the dynamics of the eigenvalues when the parameter $\varepsilon$ changes near zero, in the middle zone and near the infinity. We present several results for general potentials and some classes of analytic ones and present models when the problem can be solved explicitly. We will discuss the applications of the obtained results to the celebrated Orr–Sommerfeld problem in hydrodynamics. The talk is based on the joint papers with S. N. Tumanov.