Abstract
The Schrödinger 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 \( {Q}_1(x)=\frac{1}{4}{h}^2{\mathrm{e}}^{\hbox{--} 4ix} + 2{h}^2 \cos\ 2x \) are studied using their realizations in two-dimensional conformal field theory (2dCFT). It is shown that for the weak coupling (small) h ∈ ℝ and non-integer Floquet parameter ν ∉ ℤ spectra of hamiltonians ℋi = − d2/dx 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 nonhermitian hamiltonians which are PT-symmetric (= invariant under simultaneous parity P and time reversal T transformations) can have real eigenvalues. The hamiltonian ℋ1 is PT-symmetric for h, x ∈ ℝ. It is found that ℋ1 has a real spectrum in the weak coupling region for ν ∈ ℝ\ℤ. This fact in an elementary way follows from a definition of the N f = 1 classical irregular block. Thus, ℋ1 can serve as yet another new model for testing postulates of PT-symmetric quantum mechanics.
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Piatek, M., Pietrykowski, A.R. Classical irregular blocks, Hill’s equation and PT-symmetric periodic complex potentials. J. High Energ. Phys. 2016, 131 (2016). https://doi.org/10.1007/JHEP07(2016)131
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DOI: https://doi.org/10.1007/JHEP07(2016)131