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

Abstract

The Schroedinger eigenvalue problems for the Whittaker-Hill potential $Q_{2}(x)=\tfrac{1}{2} h^2\cos4x+4h\mu\cos2x$ and the periodic complex potential $Q_{1}(x)=\tfrac{1}{4}h^2{\rm e}^{-4ix}+2h^2\cos2x$ 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 $\nu\notin\mathbb{Z}$ spectra of hamiltonians $H_{i}\!=\!-{\rm d}^2/{\rm d}x^2 + Q_{i}(x)$, $i=1,2$ 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.