A generalized family of discrete PT-symmetric square wells

Abstract

N-site-lattice Hamiltonians H are introduced and perceived as a set of systematic discrete approximants of a certain 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 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.

21 pages, 2 figures

References in corpus (3)

• Making Sense of Non-Hermitian Hamiltonians
• On the metric operator for the imaginary cubic oscillator
• Closed formula for the metric in the Hilbert space of a PT-symmetric model

Cited by in corpus (7)

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• Three solvable matrix models of a quantum catastrophe
• Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding
• Hilbert space inner products for PT-symmetric Su-Schrieffer-Heeger models
• Quantization of Big Bang in crypto-Hermitian Heisenberg picture
• Mathematical and physical meaning of the crossings of energy levels in $\mathcal{P}\mathcal{T}-$symmetric systems
• New Concept of Solvability in Quantum Mechanics