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

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

2006-01-09

arxiv.org/abs/quant-ph/0601048v1

Phys. Lett. A 353 (2006) 463 - 468

Physics

Quantum Physics

10 pp. + 1 figure

Scientific paper

10.1016/j.physleta.2006.01.014

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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