Classical irregular blocks, Hill's equation and PT-symmetric periodic complex potentials

The Schroedinger eigenvalue problems for the Whittaker-Hill potential $Q_2(x)=\frac{1}{2}{h}^2\cos 4x+4h\mu \cos 2x$ and the periodic complex potential $Q_1(x)=\frac{1}{4}{h}^2{\mathrm{e}}^{-4ix}+2h^2\cos 2x$ are studied using their realizations in two-dimensional conformal field theory (2dCFT). It is shown that for the weak coupling (small) $h\in \mathbb{R}$ and non-integer Floquet parameter $\nu \mathrm{\notin }\mathbb{Z}$ spectra of hamiltonians $H_i\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}-{\mathrm{d}}^2\mathrm{/}\mathrm{d}{x}^2+Q_i(x)$, $i=1,2$ and corresponding two linearly independent eigenfunctions are given by the classical limit of the “single flavor” and “two flavors” ($N_f=1,2$) irregular conformal blocks. It is known that complex non-hermitian hamiltonians which are PT-symmetric (= invariant under simultaneous parity P and time reversal T transformations) can have real eigenvalues. The hamiltonian $H_1$ is PT-symmetric for $h,x\in \mathbb{R}$. It is found that $H_1$ has a real spectrum in the weak coupling region for $\nu \in \mathbb{R}\setminus \mathbb{Z}$. This fact in an elementary way follows from a definition of the $N_f=1$ classical irregular block. Thus, $H_1$ can serve as yet another new model for testing postulates of PT-symmetric quantum mechanics.