Gegenbauer-solvable quantum chain model

Abstract

In an innovative inverse-problem construction the measured, experimental energies $E_1$, $E_2$, ...$E_N$ of a quantum bound-state system are assumed fitted by an N-plet of zeros of a classical orthogonal polynomial $f_N(E)$. We reconstruct the underlying Hamiltonian $H$ (in the most elementary nearest-neighbor-interaction form) and the underlying Hilbert space $\mathcal{H}$ of states (the rich menu of non-equivalent inner products is offered). The Gegenbauer's ultraspherical polynomials $f_n(x)=C_n^{\alpha }(x)$ are chosen for the detailed illustration of technicalities.

29 pp., 1 fig

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Cited by in corpus (41)

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• Degrees and signatures of broken PT-symmetry in (non-uniform) lattices
• PT-restoration via increased loss-gain in PT-symmetric Aubry-Andre model
• Non-Hermitian anisotropic XY model with intrinsic rotation-time reversal symmetry
• Origin of maximal symmetry breaking in even PT-symmetric lattices
• Self-sustained emission in semi-infinite non-Hermitian systems at the exceptional point
• Hermitian dynamics in a class of pseudo-Hermitian networks
• Probability-preserving evolution in a non-Hermitian two-band model
• PT-symmetry breaking and universal chirality in a PT-symmetric ring
• Scaling behavior and phase diagram of a PT-symmetric non-Hermitian Bose-Hubbard system
• Simple harmonic oscillation in non-Hermitian SSH chain at exceptional point
• An exactly solvable quantum-lattice model with a tunable degree of nonlocality
• Transport properties in a Non-Hermitian triple-quantum-dot structure
• Interplay between Fano resonance and $\mathcal{P}\mathcal{T}$ symmetry in non-Hermitian discrete systems
• Admissible perturbations and false instabilities in PT-symmetric quantum systems
• Long-range entangled zero-mode state in a non-Hermitian lattice
• Discrete quantum square well of the first kind
• Complex symmetric Hamiltonians and exceptional points of order four and five
• Non linear pseudo-bosons versus hidden Hermiticity. II: The case of unbounded operators
• Exceptional point induced lasing dynamics in a non-Hermitian system
• Competing PT potentials and re-entrant PT symmetric phase for a particle in a box
• Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding
• Quantum inner-product metrics via recurrent solution of Dieudonne equation
• Non linear pseudo-bosons versus hidden Hermiticity
• PT-symmetry breaking with divergent potentials: lattice and continuum cases
• Veiled symmetry of disordered Parity-Time lattices: protected PT-threshold and the fate of localization
• Wide-range-tunable Dirac-cone band structure in a chiral-time-symmetric non-Hermitian system
• Exactly solvable $\mathcal{P}\mathcal{T}$-symmetric models in two dimensions
• Solvable non-Hermitian discrete square well with closed-form physical inner product
• Parity-time symmetry under magnetic flux
• PT-symmetric quantum systems with positive P
• On a class of non-Hermitian Hamiltonians with tridiagonal matrix representation
• PT symmetric lattices with a local degree of freedom
• PT-symmetry breaking and maximal chirality in a nonuniform PT-symmetric ring
• Semi-localization transition driven by a single asymmetrical tunneling
• CPT-symmetric discrete square well
• Parallel dynamics between non-Hermitian and Hermitian systems
• Exceptional spectrum and dynamic magnetization
• PT-symmetric quantum models living in an auxiliary Pontryagin space