  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.
