Strengthened PT-symmetry with P $\ne$ P${}^{†}$

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 $P=P^{†}$ 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 $S=P^{-1}{P}^{†}\ne I$ 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.

10 pp. + 1 figure

References in corpus (3)

• Fragile PT-symmetry in a solvable model
• Third order spectral branch points in Krein space related setups: PT-symmetric matrix toy model, MHD alpha^2-dynamo, and extended Squire equation
• Solvable relativistic quantum dots with vibrational spectra

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• Determination of the domain of the admissible matrix elements in the four-dimensional PT-symmetric anharmonic model
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• Quantum star-graph analogues of PT-symmetric square wells. II: Spectra
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