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## Gegenbauer-solvable quantum chain model

Miloslav Znojil

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 ${\cal H}$ 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.

http://arxiv.org/abs/1011.4803
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Phys. Rev. A 82 (2010) 052113
DOI:10.1103/PhysRevA.82.052113