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A Generalized Family of Discrete \(\mathcal{PT}\)-symmetric Square Wells

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Abstract

N-site-lattice Hamiltonians H (N) are introduced and perceived as a set of systematic discrete approximants of a certain \(\mathcal {PT}\)-symmetric square-well-potential model with the real spectrum and with a non-Hermiticity which is localized near the boundaries of the interval. Its strength is controlled by one, two or three parameters. The problem of the explicit construction of a nontrivial metric which makes the theory unitary is then addressed. It is proposed and demonstrated that due to the not too complicated (viz., tridiagonal matrix) form of our input Hamiltonians, the computation of the metric is straightforward and that its matrix elements prove obtainable, non-numerically, in elementary polynomial forms.

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Acknowledgements

Participation of MZ was supported by the GAČR grant Nr. P203/11/1433. Participation of JW was supported by the Natural Science Foundations of China (11171301) and by the Doctoral Programs Foundation of Ministry of Education of China (20120101110050).

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Authors and Affiliations

  1. Nuclear Physics Institute ASCR, 250 68, Rez, Czech Republic

    Miloslav Znojil

  2. Department of Mathematics, College of Science, Zhejiang University, Hangzhou, 310027, Zhejiang, P.R. China

    Junde Wu

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  1. Miloslav Znojil
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  2. Junde Wu
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Correspondence to Miloslav Znojil.

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Znojil, M., Wu, J. A Generalized Family of Discrete \(\mathcal{PT}\)-symmetric Square Wells. Int J Theor Phys 52, 2152–2162 (2013). https://doi.org/10.1007/s10773-013-1525-3

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  • DOI: https://doi.org/10.1007/s10773-013-1525-3

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