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arXiv:quant-ph/0511085 · Journal of Physics A Mathematical and General

Coupled-channel version of the PT-symmetric square well

Miloslav Znojil

Published Dec 14, 2005• 31 citations
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Abstract

A coupled pair of PT-symmetric square wells is studied as a prototype of a quantum system characterized by two manifestly non-Hermitian commuting observables. Via the diagonalization of our Hamiltonian H ≠ H † and spin-like observable Ω ≠ Ω†, we demonstrate that there exists a domain of couplings where both the respective sets of eigenvalues E n , n = 0, 1, ... (energies), and σ = ±1 ('spin projections') remain real . In such a 'measurable' regime, the model acquires a consistent probabilistic interpretation mediated by our selection of one of many available interaction-dependent scalar products.

DOI
10.1088/0305-4470/39/2/014

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A coupled pair of PT-symmetric square wells is studied as a prototype of a quantum system characterized by two manifestly non-Hermitian commuting observables. Via the diagonalization of our Hamiltonian H ≠ H † and spin-like observable Ω ≠ Ω†, we demonstrate that there exists a domain of couplings where both the respective sets of eigenvalues E n , n = 0, 1, ... (energies), and σ = ±1 ('spin projections') remain real . In such a 'measurable' regime, the model acquires a consistent probabilistic interpretation mediated by our selection of one of many available interaction-dependent scalar products.
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A pseudo-Hermitian coupled-channel square-well model is proposed, solved and discussed. The domain of parameters is determined where all the bound-state energies (twice degenerate with respect to the second observable which we call "spin") remain real. The standard probabilistic interpretation of the wave functions is achieved by the transition from the original elementary non-physical indefinite pseudo-metric θθθ to another, more involved but correct positive-definite physical metric ΘΘΘ. In our model the latter step remains comparatively easily feasible. Another fortunate circumstance emerges during the perturbative construction of the metric ΘΘΘ and, hence, of the correct scalar product and of the physical norm. One finds that at the higher energy levels the influence of all the non-Hermiticities of our model becomes very strongly and progressively suppressed.
A discrete N-point Runge–Kutta version H(N)(λ) of one of the simplest non-Hermitian square-well Hamiltonians with real spectrum is studied. Its possible Hermitizations mediated by nontrivial (often called “non-Dirac”) metrics Θ≠I are considered as a source of nonequivalent standard probabilistic interpretations of this quantum model. A complete set of these alternative, multiparametric metrics Θ=Θ(a,b,…)(N)(λ) defining all the eligible Hamiltonian-dependent representations of the physical Hilbert space of states is constructed, in closed form, for any coupling λ∊(−1,1) and for any matrix dimension N.
Bound states generated by K coupled PT-symmetric square wells are studied in a series of models where the Hamiltonians are assumed R−R-R−pseudo-Hermitian and R2−R^2-R2−symmetric. Specific rotation-like generalized parities RRR are considered such that RN=IR^N=IRN=I at some integers N. We show that and how our assumptions make the models exactly solvable and quasi-Hermitian. This means that they possess the real spectra as well as the standard probabilistic interpretation.
During recent developments in quantum theory it has been clarified that observable quantities (such as energy and position) may be represented by operators $\mathrm{\ensuremath{\Lambda}}$ (with real spectra) that are manifestly non-Hermitian in a preselected friendly Hilbert space H(F){\mathcal{H}}^{(F)}H(F). The consistency of these models is known to require an upgrade of the inner product, i.e., mathematically speaking, a transition ${\mathcal{H}}^{(F)}\ensuremath{\rightarrow}{\mathcal{H}}^{(S)}$ to another, standard, Hilbert space. We prove that whenever we are given more than one candidate for an observable (i.e., say, two operators ${\mathrm{\ensuremath{\Lambda}}}_{0}$ and ${\mathrm{\ensuremath{\Lambda}}}_{1}$) in advance, such an upgrade need not exist in general.