Gegenbauer-solvable quantum chain model

TL;DR

This paper introduces a method to reconstruct quantum Hamiltonians and Hilbert spaces from experimental energy measurements using Gegenbauer polynomials, providing a novel inverse problem approach in quantum modeling.

Contribution

It presents a new inverse-problem framework that reconstructs quantum Hamiltonians and Hilbert spaces from energy data using orthogonal polynomial zeros, specifically Gegenbauer polynomials.

Findings

Reconstruction of Hamiltonian from energy eigenvalues.

Application of Gegenbauer polynomials for detailed illustration.

Framework for defining the underlying Hilbert space structure.

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 ${\cal H}$ 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.