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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• An exactly solvable quantum-lattice model with a tunable degree of nonlocality
• Discrete quantum square well of the first kind
• CPT-symmetric discrete square well