Matching method and exact solvability of discrete PT-symmetric square wells

Abstract

Discrete PT-symmetric square wells are studied. Their wave functions are found proportional to classical Tshebyshev polynomials of complex argument. The compact secular equations for energies are derived giving the real spectra in certain intervals of non-Hermiticity strengths Z. It is amusing to notice that although the known square well re-emerges in the usual continuum limit, a twice as rich, upside-down symmetric spectrum is exhibited by all its present discretized predecessors.

25 pp, 3 figures

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• Confluences of exceptional points and a systematic classification of quantum catastrophes
• N-site-lattice analogues of $V(x)=i{x}^3$
• Solvable quantum lattices with nonlocal non-Hermitian endpoint interactions
• Bound states emerging from below the continuum in a solvable PT-symmetric discrete Schroedinger equation
• On the Spectrum of a Discrete Non-Hermitian Quantum System
• CPT-symmetric discrete square well
• Double complex SUSY-transformations: deformations of real potentials and their spectral characteristics
• Cryptohermitian Hamiltonians on graphs. II. Hermitizations