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Solvable non-Hermitian discrete square well with closed-form physical inner product. (English) Zbl 1315.81044

From the author’s abstract: A new Hermitizable quantum model is proposed in which the bound-state energies are real and given as roots of an elementary trigonometric expression while the wave function components are expressed as superpositions of two Chebyshev polynomials. As an \(N\)-site lattice version of square well with complex Robin-type two-parametric boundary conditions the model is unitary with respect to the Hilbert space metric \(\Theta\) which becomes equal to the most common Dirac’s metric \(\Theta^{\text{(Dirac)}}=I\) in the conventional textbook Hermitian-Hamiltonian limit. This metric is constructed in closed form at all \(N=2,3,\dots \).

MSC:

81Q12 Nonselfadjoint operator theory in quantum theory including creation and destruction operators
81Q80 Special quantum systems, such as solvable systems
41A50 Best approximation, Chebyshev systems
39A12 Discrete version of topics in analysis
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