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Strengthened PT-symmetry with P $\neq$ P$^\dagger$

Miloslav Znojil
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Abstract

Two alternative scenarios are shown possible in Quantum Mechanics working with non-Hermitian $PT-$symmetric form of observables. While, usually, people assume that $P$ is a self-adjoint indefinite metric in Hilbert space (and that their $P-$pseudo-Hermitian Hamiltonians $H$ possess the real spectra etc), we propose to relax the constraint $P=P^\dagger$ as redundant. Non-Hermitian triplet of coupled square wells is chosen for illustration purposes. Its solutions are constructed and the observed degeneracy of their spectrum is attributed to the characteristic nontrivial symmetry $S={P}^{-1} {P}^\dagger \neq I$ of the model $H$. Due to the solvability of the model the determination of the domain where the energies remain real is straightforward. A few remarks on the correct (albeit ambiguous) physical interpretation of the model are added.

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BibTeXRIS

Miloslav Znojil. 2006-01-09. Strengthened PT-symmetry with P $\neq$ P$^\dagger$. https://doi.org/10.1016/j.physleta.2006.01.014

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