Conjecture on the Analyticity of PT-Symmetric Potentials and the Reality of their Spectra
Physics and Astronomy
The spectrum of the Hermitian Hamiltonian H = p(2) + V (x) is real and discrete if the potential V (x) -> infinity as x -> +/-infinity. However, if V (x) is complex and PT-symmetric, it is conjectured that, except in rare special cases, V (x) must be analytic in order to have a real spectrum. This conjecture is demonstrated by using the potential V (x) = (ix)(a)|x|(b), where a, b are real.
Journal of Physics A-Mathematical and Theoretical
Bender, C. M.,
Hook, D. W.,
Mead, L. R.
(2008). Conjecture on the Analyticity of PT-Symmetric Potentials and the Reality of their Spectra. Journal of Physics A-Mathematical and Theoretical, 41(39).
Available at: https://aquila.usm.edu/fac_pubs/1521