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

Abstract

The Schroedinger eigenvalue problems for the Whittaker-Hill potential $Q_2(x)=\frac{1}{2}{h}^2\cos 4x+4h\mu \cos 2x$ 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 $\nu \notin \mathbb{Z}$ spectra of hamiltonians $H_i=-{\mathrm{d}}^2/\mathrm{d}{x}^2+Q_i(x)$, $i=1,2$ 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.

23 pages, 1 figure

References in corpus (7)

• Making Sense of Non-Hermitian Hamiltonians
• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems
• Classical irregular block, N=2 pure gauge theory and Mathieu equation
• Classical Virasoro irregular conformal block
• Classical Virasoro irregular conformal block II
• Classical limit of irregular blocks and Mathieu functions

Cited by in corpus (7)

• A slow review of the AGT correspondence
• Accessory parameters in confluent Heun equations and classical irregular conformal blocks
• Quantum periods and prepotential in $\mathcal{N}=2$ SU(2) SQCD
• Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models
• Cooling in a parametrically driven optomechanical cavity
• Irregular Conformal States and Spectral Curve: Irregular Matrix Model Approach
• Solvable spectral problems from 2d CFT and N=2 gauge theories