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

References in corpus (4)

• MHD alpha^2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator
• Fragile PT-symmetry in a solvable model
• Third order spectral branch points in Krein space related setups: PT-symmetric matrix toy model, MHD alpha^2-dynamo, and extended Squire equation
• Solvable relativistic quantum dots with vibrational spectra

Cited by in corpus (10)

• Scattering theory with localized non-Hermiticities
• Maximal couplings in PT-symmetric chain-models with the real spectrum of energies
• Discrete PT-symmetric models of scattering
• Scattering theory using smeared non-Hermitian potentials
• A return to observability near exceptional points in a schematic PT-symmetric model
• Fundamental length in quantum theories with PT-symmetric Hamiltonians
• Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs
• Inverse Spectral Problems for Tridiagonal N by N Complex Hamiltonians
• On the Spectrum of a Discrete Non-Hermitian Quantum System
• Double complex SUSY-transformations: deformations of real potentials and their spectral characteristics