Nonlinearity of perturbations in PT-symmetric quantum mechanics

0509984 - ÚJF 2020 RIV GB eng C - Konferenční příspěvek (zahraniční konf.)
Znojil, Miloslav - Růžička, František
Nonlinearity of perturbations in PT-symmetric quantum mechanics.
Journal of Physics: Conference series. Vol. 1194. Bristol: IOP, 2019 - (Burdík, Č.; Navrátil, O.; Pošta, S.), č. článku 012120. ISSN 1742-6588.
[32nd International Colloquium on Group Theoretical Methods in Physics (Group32). Prague (CZ), 08.07.2018-13.07.2018]
Grant CEP: GA ČR GA16-22945S
Institucionální podpora: RVO:61389005
Klíčová slova: Group theory * Hilbert spaces * quantum optics
Obor OECD: Atomic, molecular and chemical physics (physics of atoms and molecules including collision, interaction with radiation, magnetic resonances, Mössbauer effect)

In the so called crypto-Hermitian formulation of quantum theory (incorporating, in particular, the PT -symmetric quantum mechanics as its special case) the unitary evolution of a system is known to be described via an apparently redundant representation of the states in a triplet of Hilbert spaces. Two of them are unitarily equivalent while the auxiliary, zeroth one is unphysical but exceptionally user-friendly. The dynamical evolution equations are, naturally, solved in the friendliest space H(0). The evaluation of experimental predictions then requires a Hermitization of the observables. This yields, as its byproduct, the correct physical Hilbert space H(1). The formalism offers the conventional probabilistic interpretation due to the Dyson-proposed unitary equivalence between H(1)and a certain conventional but prohibitively complicated textbook space H(2). The key merit of the innovation lies in its enhanced flexibility opening new ways towards nonlocal or complex-interaction unitary models. The price to pay is that the ad hoc inner product in the relevant physical Hilbert space H(1) is, by construction, Hamiltonian-dependent. We show that this implies that the Rayleigh-Schrödinger perturbation theory must be used with due care. The warning is supported by an elementary sample of quantum system near its exceptional-point phase transition. In contrast to a naive expectation, the model proves stable with respect to the admissible, self-consistently specified (i.e., physical) small perturbations.
Trvalý link: http://hdl.handle.net/11104/0300554